Apeirostikìc Logismìc II

Apeirostikìc Logismìc II Prìqeirec Shmei seic Tm m Mjhmtik n Pnepist mio Ajhn n -

Perieqìmen UpkoloujÐec ki bsikèc koloujðec. UpkoloujÐec. Je rhm Bolzno-Weierstrss.þ Apìdeixh me qr sh thc rq c tou kibwtismoô 3.3 An tero ki kt tero ìrio koloujðc 4.4 AkoloujÐec Cuchy 8.5 *Prˆrthm: suz thsh gi to xðwm thc plhrìthtc.6 Ask seic Seirèc prgmtik n rijm n 5. SÔgklish seirˆc 5. Seirèc me mh rnhtikoôc ìrouc 9.þ Seirèc me fjðnontec mh rnhtikoôc ìrouc.bþ O rijmìc e.3 Genikˆ krit ri 4.3þ Apìluth sôgklish seirˆc 4.3bþ Krit ri sôgkrishc 5.3gþ Krit rio lìgou ki krit rio rðzc 7.3dþ To krit rio tou Dirichlet 9.3eþ *Dekdik prˆstsh prgmtik n rijm n 3.4 Dunmoseirèc 35.5 Ask seic 37 3 Omoiìmorfh sunèqei 43 3. Omoiìmorfh sunèqei 43 3. Qrkthrismìc thc omoiìmorfhc sunèqeic mèsw kolouji n 46 3.3 SuneqeÐc sunrt seic se kleistˆ dist mt 48 3.4 Sustolèc je rhm stjeroô shmeðou 5 3.5 Ask seic 5 4 Olokl rwm Riemnn 55 4. O orismìc tou Drboux 55 4. To krit rio oloklhrwsimìthtc tou Riemnn 58 4.3 DÔo klˆseic Riemnn oloklhr simwn sunrt sewn 63 4.4 Idiìthtec tou oloklhr mtoc Riemnn 65 4.5 O orismìc tou Riemnn* 7 4.6 Ask seic 75

iv Perieqìmen 5 To jemeli dec je rhm tou ApeirostikoÔ LogismoÔ 8 5. To je rhm mèshc tou OloklhrwtikoÔ LogismoÔ 8 5. T jemeli dh jewr mt tou ApeirostikoÔ LogismoÔ 8 5.3 Mèjodoi olokl rwshc 86 5.4 Genikeumèn oloklhr mt 88 5.4þ To krit rio tou oloklhr mtoc 9 5.5 Ask seic 9 6 Teqnikèc olokl rwshc 95 6. Olokl rwsh me ntiktˆstsh 95 6.þ PÐnkc stoiqeiwd n oloklhrwmˆtwn 95 6.bþ Upologismìc tou f(φ(x))φ (x) dx 95 6.gþ Trigwnometrikˆ oloklhr mt 96 6.dþ Upologismìc tou f(x) dx me thn ntiktˆstsh x = φ(t) 98 6. Olokl rwsh ktˆ mèrh 99 6.3 Olokl rwsh rht n sunrt sewn 6.4 Kˆpoiec qr simec ntiktstˆseic 4 6.4þ Rhtèc sunrt seic twn cos x ki sin x 4 6.4bþ Oloklhr mt lgebrik n sunrt sewn eidik c morf c 5 6.5 Ask seic 7 7 Je rhm Tylor 7. Je rhm Tylor 7. Dunmoseirèc ki nptôgmt Tylor 6 7.þ H ekjetik sunˆrthsh f(x) = e x 6 7.bþ H sunˆrthsh f(x) = cos x 6 7.gþ H sunˆrthsh f(x) = sin x 7 7.dþ H sunˆrthsh f(x) = ln( + x), x (, ] 8 7.eþ H diwnumik sunˆrthsh f(x) = ( + x), x > 9 7. þ H sunˆrthsh f(x) = rctn x, x 7.3 Sunrt seic prstˆsimec se dunmoseirˆ 7.4 Ask seic 5 8 Kurtèc ki koðlec sunrt seic 7 8. Orismìc 7 8. Kurtèc sunrt seic orismènec se noiktì diˆsthm 8 8.3 PrgwgÐsimec kurtèc sunrt seic 3 8.4 Anisìtht tou Jensen 3 8.5 Ask seic 34

Kefˆlio UpkoloujÐec ki bsikèc koloujðec. UpkoloujÐec Orismìc... 'Estw ( n ) mi koloujð prgmtik n rijm n. H koloujð (b n ) lègeti upkoloujð thc ( n ) n upˆrqei gnhsðwc Ôxous koloujð fusik n rijm n k < k < < k n < k n+ < ste (..) b n = kn gi kˆje n N. Me ˆll lìgi, oi ìroi thc (b n ) eðni oi k, k,..., kn,..., ìpou k < k < < k n < k n+ <. Genikˆ, mi koloujð èqei pollèc (sun jwc ˆpeirec to pl joc) diforetikèc upkoloujðec. PrdeÐgmt... 'Estw ( n ) mi koloujð prgmtik n rijm n. () H upkoloujð ( n ) twn {ˆrtiwn ìrwn} thc ( n ) èqei ìrouc touc, 4, 6,.... Ed, k n = n. (b) H upkoloujð ( n ) twn {peritt n ìrwn} thc ( n ) èqei ìrouc touc, 3, 5,.... Ed, k n = n. (g) H upkoloujð ( n ) thc ( n ) èqei ìrouc touc, 4, 9,.... Ed, k n = n. (d) Kˆje telikì tm m ( m, m+, m+,...) thc ( n ) eðni upkoloujð thc ( n ). Ed, k n = m + n. Prt rhsh..3. 'Estw (k n ) mi gnhsðwc Ôxous koloujð fusik n rijm n. Tìte, k n n gi kˆje n N.

UpkoloujÐes ki bsikès koloujðes Apìdeixh. Me epgwg : foô o k eðni fusikìc rijmìc, eðni fnerì ìti k. Gi to epgwgikì b m upojètoume ìti k m m. AfoÔ h (k n ) eðni gnhsðwc Ôxous, èqoume k m+ > k m, ˆr k m+ > m. AfoÔ oi k m+ ki m eðni fusikoð rijmoð, èpeti ìti k m+ m + (jumhjeðte ìti nˆmes ston m ki ston m + den upˆrqei ˆlloc fusikìc). H epìmenh Prìtsh deðqnei ìti n mi koloujð sugklðnei se prgmtikì rijmì tìte ìlec oi upkoloujðec thc eðni sugklðnousec ki sugklðnoun ston Ðdio prgmtikì rijmì. Prìtsh..4. An n tìte gi kˆje upkoloujð ( kn ) thc ( n ) isqôei kn. Apìdeixh. 'Estw ε >. AfoÔ n, upˆrqei n = n (ε) N me thn ex c idiìtht: Gi kˆje m n isqôei m < ε. Apì thn Prt rhsh..3 gi kˆje n n èqoume k n n n. Jètontc loipìn m = k n sthn prohgoômenh sqèsh, pðrnoume: Gi kˆje n n isqôei kn < ε. Autì podeiknôei ìti kn : gi to tuqìn ε > br kme n N ste ìloi oi ìroi kn,,... thc ( kn + k n ) n n koun sto ( ε, + ε). Prt rhsh..5. H prohgoômenh Prìtsh eðni polô qr simh n jèloume n deðxoume ìti mi koloujð ( n ) den sugklðnei se knènn prgmtikì rijmì. ArkeÐ n broôme dôo upkoloujðec thc ( n ) oi opoðec n èqoun diforetikˆ ìri. Gi prˆdeigm, c jewr soume thn ( n ) = ( ) n. Tìte, n = ( ) n = ki n = ( ) n =. Ac upojèsoume ìti n. Oi ( n ) ki ( n ) eðni upkoloujðec thc ( n ), prèpei loipìn n isqôei n ki n. Apì th mondikìtht tou orðou thc ( n ) pðrnoume = ki pì th mondikìtht tou orðou thc ( n ) pðrnoume =. Dhld, =. Ktl xme se ˆtopo, ˆr h ( n ) den sugklðnei.. Je rhm Bolzno-Weierstrss Je rhm.. (Bolzno-Weierstrss). Kˆje frgmènh koloujð èqei toulˆqiston mð upkoloujð pou sugklðnei se prgmtikì rijmì. J d soume dôo podeðxeic utoô tou Jewr mtoc. H pr th bsðzeti sto gegonìc ìti kˆje monìtonh ki frgmènh koloujð sugklðnei. Gi n broôme sugklðnous upkoloujð mic frgmènhc koloujðc rkeð n broôme mi monìtonh upkoloujð thc. To teleutðo isqôei entel c genikˆ, ìpwc deðqnei to epìmeno Je rhm: Je rhm... Kˆje koloujð èqei toulˆqiston mð monìtonh upkoloujð. Apìdeixh. J qreistoôme thn ènnoi tou shmeðou koruf c mic koloujðc. Orismìc..3. 'Estw ( n ) mi koloujð prgmtik n rijm n. Lème ìti o m eðni shmeðo koruf c thc ( n ) n m n gi kˆje n m. [Gi n exoikeiwjeðte me ton orismì elègxte t ex c. An h ( n ) eðni fjðnous tìte kˆje ìroc thc eðni shmeðo koruf c thc. An h ( n ) eðni gnhsðwc Ôxous tìte den èqei knèn shmeðo koruf c.]

. Je rhm Bolzno-Weierstrss 3 'Estw ( n ) mi koloujð prgmtik n rijm n. DikrÐnoume dôo peript seic: () H ( n ) èqei ˆpeir to pl joc shmeð koruf c. Tìte, upˆrqoun fusikoð rijmoð k < k < < k n < k n+ < ste ìloi oi ìroi k,..., kn,... n eðni shmeð koruf c thc ( n ) (exhg ste gitð). AfoÔ k n < k n+ gi kˆje n N, h ( kn ) eðni upkoloujð thc ( n ). Apì ton orismì tou shmeðou koruf c blèpoume ìti gi kˆje n N isqôei kn kn+ (èqoume k n+ > k n ki o kn eðni shmeðo koruf c thc ( n )). Dhld, (..) k k kn kn+. 'Ar, h upkoloujð ( kn ) eðni fjðnous. (b) H ( n ) èqei pepersmèn to pl joc shmeð koruf c. Tìte, upˆrqei N N me thn ex c idiìtht: n m N tìte o m den eðni shmeðo koruf c thc ( n ) (pˆrte N = k + ìpou k to teleutðo shmeðo koruf c thc ( n ) N = n den upˆrqoun shmeð koruf c). Me bˆsh ton orismì tou shmeðou koruf c utì shmðnei ìti: n m N tìte upˆrqei n > m ste n > m. Efrmìzoume didoqikˆ to prpˆnw. Jètoume k = N ki brðskoume k > k ste k > k. Ktìpin brðskoume k 3 > k ste k3 > k ki oôtw kjex c. Upˆrqoun dhld k < k < < k n < k n+ < ste (..) k < k < < kn < kn+ <. Tìte, h ( kn ) eðni gnhsðwc Ôxous upkoloujð thc ( n ). MporoÔme t r n podeðxoume to Je rhm Bolzno-Weierstrss. Apìdeixh tou Jewr mtoc... 'Estw ( n ) frgmènh koloujð. Apì to Je rhm.. h ( n ) èqei monìtonh upkoloujð ( kn ). H ( kn ) eðni monìtonh ki frgmènh, sunep c sugklðnei se prgmtikì rijmì..þ Apìdeixh me qr sh thc rq c tou kibwtismoô H deôterh pìdeixh tou Jewr mtoc Bolzno-Weierstrss qrhsimopoieð thn rq twn kibwtismènwn disthmˆtwn. 'Estw ( n ) mi frgmènh koloujð prgmtik n rijm n. Tìte, upˆrqei kleistì diˆsthm [b, c ] sto opoðo n koun ìloi oi ìroi n. QwrÐzoume to [b, c ] se dôo didoqikˆ dist mt pou èqoun to Ðdio m koc c b : t [ ] [ b, b+c ki b+c ], c. Kˆpoio pì utˆ t dôo dist mt perièqei ˆpeirouc to pl joc ìrouc thc ( n ). PÐrnontc sn [b, c ] utì to upodiˆsthm tou [b, c ] èqoume deðxei to ex c. Upˆrqei kleistì diˆsthm [b, c ] [b, c ] to opoðo perièqei ˆpeirouc ìrouc thc ( n ) ki èqei m koc (..3) c b = c b. SuneqÐzoume me ton Ðdio trìpo: qwrðzoume to [b, c ] se dôo didoqikˆ dist mt m kouc c b : t [ ] [ b, b+c ki b+c ], c. AfoÔ to [b, c ] perièqei ˆpeirouc ìrouc thc ( n ), kˆpoio pì utˆ t dôo dist mt perièqei ˆpeirouc to pl joc ìrouc thc ( n ). PÐrnontc sn [b 3, c 3 ] utì to upodiˆsthm tou [b, c ] èqoume deðxei to ex c.

4 UpkoloujÐes ki bsikès koloujðes Upˆrqei kleistì diˆsthm [b 3, c 3 ] [b, c ] to opoðo perièqei ˆpeirouc ìrouc thc ( n ) ki èqei m koc (..4) c 3 b 3 = c b = c b. SuneqÐzontc me ton Ðdio trìpo orðzoume koloujð ( [b m, c m ] ) kleist n disthmˆtwn pou iknopoieð t ex m N c: (i) Gi kˆje m N isqôei [b m+, c m+ ] [b m, c m ]. (ii) Gi kˆje m N isqôei c m b m = (c b )/ m. (iii) Gi kˆje m N upˆrqoun ˆpeiroi ìroi thc ( n ) sto [b m, c m ]. Qrhsimopoi ntc thn trðth sunj kh, mporoôme n broôme upkoloujð ( km ) thc ( n ) me thn idiìtht: gi kˆje m N isqôei km [b m, c m ]. Prˆgmti, upˆrqei k N ste k [b, c ] gi thn krðbei, ìloi oi ìroi thc ( n ) brðskonti sto [b, c ]. T r, foô to [b, c ] perièqei ˆpeirouc ìrouc thc ( n ), kˆpoioc pì utoôc èqei deðkth meglôtero pì k. Dhld, upˆrqei k > k ste k [b, c ]. Me ton Ðdio trìpo, n èqoun oristeð k < < k m ste ks [b s, c s ] gi kˆje s =,..., m, mporoôme n broôme k m+ > k m ste km+ [b m+, c m+ ] (diìti, to [b m+, c m+ ] perièqei ˆpeirouc ìrouc thc ( n )). 'Etsi, orðzeti mi upkoloujð ( km ) thc ( n ) pou iknopoieð to zhtoômeno. J deðxoume ìti h ( km ) sugklðnei. Apì thn rq twn kibwtismènwn disthmˆtwn (ki lìgw thc (ii)) upˆrqei mondikìc R o opoðoc n kei se ìl t kleistˆ dist mt [b m, c m ]. JumhjeÐte ìti AfoÔ b m km lim b m = = lim c m. m m c m gi kˆje m, to krit rio twn isosugklinous n kolouji n deðqnei ìti km..3 An tero ki kt tero ìrio koloujðc Skopìc mc se ut n thn Prˆgrfo eðni n melet soume pio prosektikˆ tic upkoloujðec mic frgmènhc koloujðc. JumhjeÐte ìti n h koloujð ( n ) sugklðnei se kˆpoion prgmtikì rijmì tìte h ktˆstsh eðni polô pl. An ( kn ) eðni tuqoôs upkoloujð thc ( n ), tìte kn. Dhld, ìlec oi upkoloujðec mic sugklðnousc koloujðc sugklðnoun ki mˆlist sto ìrio thc koloujðc. Orismìc.3.. 'Estw ( n ) mi koloujð. Lème ìti o x R eðni orikì shmeðo ( upkoloujikì ìrio) thc ( n ) n upˆrqei upkoloujð ( kn ) thc ( n ) ste kn x. T orikˆ shmeð mic koloujðc qrkthrðzonti pì to epìmeno L mm. L mm.3.. O x eðni orikì shmeðo thc ( n ) n ki mìno n gi kˆje ε > ki gi kˆje m N upˆrqei n m ste n x < ε. Apìdeixh. Upojètoume pr t ìti o x eðni orikì shmeðo thc ( n ). Upˆrqei loipìn upkoloujð ( kn ) thc ( n ) ste kn x.

.3 An tero ki kt tero ìrio koloujðs 5 'Estw ε > ki m N. Upˆrqei n N ste kn x < ε gi kˆje n n. JewroÔme ton n = mx{m, n }. Tìte k n n m ki n n, ˆr kn x < ε. AntÐstrof: PÐrnoume ε = ki m =. Apì thn upìjesh upˆrqei k ste k x <. Sth sunèqei pðrnoume ε = ki m = k +. Efrmìzontc thn upìjesh brðskoume k k + > k ste k x <. Epgwgikˆ brðskoume k < k < < k n < ste kn x < n (kˆnete mìnoi sc to epgwgikì b m). EÐni fnerì ìti kn x. 'Estw ( n ) mi frgmènh koloujð. Dhld, upˆrqei M > ste n M gi kˆje n N. JewroÔme to sônolo (.3.) K = {x R : x eðni orikì shmeðo thc ( n )}.. To K eðni mh kenì. Apì to Je rhm Bolzno-Weierstrss upˆrqei toulˆqiston mð upkoloujð ( kn ) thc ( n ) pou sugklðnei se prgmtikì rijmì. To ìrio thc ( kn ) eðni ex orismoô stoiqeðo tou K.. To K eðni frgmèno. An x K, upˆrqei kn x ki foô M kn M gi kˆje n, èpeti ìti M x M. Apì to xðwm thc plhrìthtc prokôptei ìti upˆrqoun t sup K ki inf K. epìmeno L mm deðqnei ìti to K èqei mègisto ki elˆqisto stoiqeðo. L mm.3.3. 'Estw ( n ) frgmènh koloujð ki Tìte, sup K K ki inf K K. K = {x R : x eðni orikì shmeðo thc ( n )}. Apìdeixh. 'Estw = sup K. Jèloume n deðxoume ìti o eðni orikì shmeðo thc ( n ), ki sômfwn me to L mm.3. rkeð n doôme ìti gi kˆje ε > ki gi kˆje m N upˆrqei n m ste n < ε. 'Estw ε > ki m N. AfoÔ = sup K, upˆrqei x K ste ε < x. O x eðni orikì shmeðo thc ( n ), ˆr upˆrqei n m ste n x < ε. Tìte, (.3.) n n x + x < ε + ε = ε. To Me nˆlogo trìpo deðqnoume ìti inf K K. Orismìc.3.4. 'Estw ( n ) mi frgmènh koloujð. An K = {x R : x eðni orikì shmeðo thc ( n )}, orðzoume (i) lim sup n = sup K, to n tero ìrio thc ( n ), (ii) lim inf n = inf K to kt tero ìrio thc ( n ). SÔmfwn me to L mm.3.3, to lim sup n eðni to mègisto stoiqeðo ki to lim inf n eðni to elˆqisto stoiqeðo tou K ntðstoiq:

6 UpkoloujÐes ki bsikès koloujðes Je rhm.3.5. 'Estw ( n ) frgmènh koloujð. To lim sup n eðni o meglôteroc prgmtikìc rijmìc x gi ton opoðo upˆrqei upkoloujð ( kn ) thc ( n ) me kn x. To lim inf n eðni o mikrìteroc prgmtikìc rijmìc y gi ton opoðo upˆrqei upkoloujð ( ln ) thc ( n ) me ln y. To n tero ki to kt tero ìrio mic frgmènhc koloujðc perigrˆfonti mèsw twn perioq n touc wc ex c: Je rhm.3.6. 'Estw ( n ) frgmènh koloujð prgmtik n rijm n ki èstw x R. Tìte, () x lim sup n n ki mìno n: gi kˆje ε > to sônolo {n N : x ε < n } eðni ˆpeiro. () x lim sup n n ki mìno n: gi kˆje ε > to sônolo {n N : x + ε < n } eðni pepersmèno. (3) x lim inf n n ki mìno n: gi kˆje ε > to sônolo {n N : n < x + ε} eðni ˆpeiro. (4) x lim inf n n ki mìno n: gi kˆje ε > to sônolo {n N : n < x ε} eðni pepersmèno. (5) x = lim sup n n ki mìno n: gi kˆje ε > to {n N : x ε < n } eðni ˆpeiro ki to {n N : x + ε < n } eðni pepersmèno. (6) x = lim inf n n ki mìno n: gi kˆje ε > to {n N : n < x + ε} eðni ˆpeiro ki to {n N : n < x ε} eðni pepersmèno. Apìdeixh. (: ) 'Estw ε >. Upˆrqei upkoloujð ( kn ) thc ( n ) me kn lim sup n, ˆr upˆrqei n ste gi kˆje n n (.3.3) kn > lim sup n ε x ε. 'Epeti ìti to {n : n > x ε} eðni ˆpeiro. (: ) 'Estw ε >. Ac upojèsoume ìti upˆrqoun k < k < < k n < me kn > x + ε. Tìte, h upkoloujð ( kn ) thc ( n ) èqei ìlouc touc ìrouc thc meglôterouc pì x + ε. MporoÔme n broôme sugklðnous upkoloujð ( ksn ) thc ( kn ) (pì to Je rhm Bolzno-Weierstrss) ki tìte ksn y x + ε. 'Omwc tìte, h ( ksn ) eðni upkoloujð thc ( n ) (exhg ste gitð), opìte (.3.4) lim sup n y x + ε lim sup n + ε. Autì eðni ˆtopo. 'Ar, to {n : n > x + ε} eðni pepersmèno. (: ) 'Estw ìti x > lim sup n. Tìte upˆrqei ε > ste n y = x ε n èqoume x > y > lim sup n. Apì thn upìjes mc, to {n N : y < n } eðni ˆpeiro. 'Omwc y > lim sup n opìte pì thn (: ) to sônolo {n N : y < n } eðni pepersmèno (grˆyte y = lim sup n + ε gi kˆpoio ε > ). Oi dôo isqurismoð èrqonti se ntðfsh. (: ) 'Omoi, upojètoume ìti x < lim sup n ki brðskoume y ste x < y < lim sup n. AfoÔ y > x, sumperðnoume ìti to {n N : y < n } eðni pepersmèno (ut eðni h upìjes mc) ki foô y < lim sup n sumperðnoume ìti to {n N : y < n } eðni ˆpeiro (pì thn (: ).) 'Etsi ktl goume se ˆtopo. H (5) eðni ˆmesh sunèpei twn () ki (). Gi tic (3), (4) ki (6) ergzìmste ìmoi. Mi enllktik perigrf twn lim sup n ki lim inf n dðneti pì to epìmeno je rhm:

.3 An tero ki kt tero ìrio koloujðs 7 Je rhm.3.7. 'Estw ( n ) frgmènh koloujð. () Jètoume b n = sup{ k : k n}. Tìte, lim sup n = inf{b n : n N}. (b) Jètoume γ n = inf{ k : k n}. Tìte, lim inf n = sup{γ n : n N}. Apìdeixh. DeÐqnoume pr t ìti oi rijmoð inf{b n : n N} ki sup{γ n : n N} orðzonti klˆ: Gi kˆje n N, isqôei γ n n b n (exhg ste gitð). EpÐshc, h (b n ) eðni fjðnous, en h ( n ) eðni Ôxous (exhg ste gitð). AfoÔ h ( n ) eðni frgmènh, èpeti ìti h (b n ) eðni fjðnous ki kˆtw frgmènh, en h (γ n ) eðni Ôxous ki ˆnw frgmènh. Apì to je rhm sôgklishc monìtonwn kolouji n sumperðnoume ìti b n inf{b n : n N} := b ki γ n sup{γ n : n N} := γ. J deðxoume ìti lim sup n = b. Apì to L mm.3.3 upˆrqei upkoloujð ( kn ) thc ( n ) me kn lim sup n. 'Omwc, kn b kn ki b kn b (exhg ste gitð). 'Ar, (.3.5) lim sup n = lim kn lim b kn = b. Gi thn ntðstrofh nisìtht deðqnoume ìti o b eðni orikì shmeðo thc ( n ). 'Estw ε > ki èstw m N. AfoÔ b n b, upˆrqei n m ste b b n < ε. Allˆ, b n = sup{ k : k n}, ˆr upˆrqei k n m ste b n k > b n ε dhld b n k < ε. 'Epeti ìti (.3.6) b k b b n + b n k < ε + ε = ε. Apì to L mm.3. o b eðni orikì shmeðo thc ( n ), ki sunep c, b lim sup n. Me nˆlogo trìpo deðqnoume ìti lim inf n = γ. KleÐnoume me ènn qrkthrismì thc sôgklishc gi frgmènec koloujðec. Je rhm.3.8. 'Estw ( n ) frgmènh koloujð. H ( n ) sugklðnei n ki mìno n lim sup n = lim inf n. Apìdeixh. An n tìte gi kˆje upkoloujð ( kn ) thc ( n ) èqoume kn. Epomènwc, o eðni to mondikì orikì shmeðo thc ( n ). 'Eqoume K = {}, ˆr lim sup n = lim inf n =. AntÐstrof: èstw ε >. Apì to Je rhm.3.6 o rijmìc = lim sup n = lim inf n èqei thn ex c idiìtht: T sônol {n N : n < ε} ki {n N : n > + ε} eðni pepersmèn. Dhld, to sônolo (.3.7) {n N : n > ε} eðni pepersmèno. IsodÔnm, upˆrqei n N me thn idiìtht: gi kˆje n n, n ε. AfoÔ to ε > tn tuqìn, èpeti ìti n.

8 UpkoloujÐes ki bsikès koloujðes Prt rhsh.3.9. Ac upojèsoume ìti h koloujð ( n ) den eðni frgmènh. An h ( n ) den eðni ˆnw frgmènh, tìte upˆrqei upkoloujð ( kn ) thc ( n ) ste kn + (ˆskhsh). Me ˆll lìgi, o + eðni {orikì shmeðo} thc ( n ). Se ut n thn perðptwsh eðni logikì n orðsoume lim sup n = +. Entel c nˆlog, n h ( n ) den eðni kˆtw frgmènh, tìte upˆrqei upkoloujð ( kn ) thc ( n ) ste kn (ˆskhsh). Dhld, o eðni {orikì shmeðo} thc ( n ). Tìte, orðzoume lim inf n =..4 AkoloujÐec Cuchy O orismìc thc koloujðc Cuchy èqei sn fethrð thn ex c prt rhsh: c upojèsoume ìti n. Tìte, oi ìroi thc ( n ) eðni telikˆ {kontˆ} sto, ˆr eðni telikˆ ki {metxô touc kontˆ}. Gi n ekfrˆsoume usthrˆ ut thn prt rhsh, c jewr soume tuqìn ε >. Upˆrqei n = n (ε) N ste gi kˆje n n n isqôei n < ε. Tìte, gi kˆje n, m n èqoume (.4.) n m n + m < ε + ε = ε. Orismìc.4.. Mi koloujð ( n ) lègeti koloujð Cuchy ( bsik koloujð) n gi kˆje ε > upˆrqei n = n (ε) N ste: (.4.) n m, n n (ε), tìte n m < ε. Prt rhsh.4.. An h ( n ) eðni koloujð Cuchy, tìte gi kˆje ε > upˆrqei n = n (ε) N ste (.4.3) n n n (ε), tìte n n+ < ε. To ntðstrofo den isqôei: n, pì kˆpoion deðkth ki pèr, didoqikoð ìroi eðni kontˆ, den èpeti ngkstikˆ ìti h koloujð eðni Cuchy. Gi prˆdeigm, jewr ste thn (.4.4) n = + + + n. Tìte, (.4.5) n+ n = ìtn n, ìmwc (.4.6) n n = n + + + n n = + n + n n ìtn n, p ìpou blèpoume ìti h ( n ) den eðni koloujð Cuchy. Prˆgmti, n h ( n ) tn koloujð Cuchy, j èprepe (efrmìzontc ton orismì me ε = ) gi megˆl n, m = n n isqôei (.4.7) n n < dhld n <, to opoðo odhgeð se ˆtopo.

.4 AkoloujÐes Cuchy 9 Skopìc mc eðni n deðxoume ìti mi koloujð prgmtik n rijm n eðni sugklðnous n ki mìno n eðni koloujð Cuchy. H pìdeixh gðneti se trð b mt. Prìtsh.4.3. Kˆje koloujð Cuchy eðni frgmènh. Apìdeixh. 'Estw ( n ) koloujð Cuchy. Pˆrte ε = > ston orismì: upˆrqei n N ste n m < gi kˆje n, m n. Eidikìter, n n < gi kˆje n > n. Dhld, (.4.8) n < + n gi kˆje n > n. Jètoume M = mx{,..., n, + n } ki eôkol eplhjeôoume ìti n M gi kˆje n N. Prìtsh.4.4. An mi koloujð Cuchy ( n ) èqei sugklðnous upkoloujð, tìte h ( n ) sugklðnei. Apìdeixh. Upojètoume ìti h ( n ) eðni koloujð Cuchy ki ìti h upkoloujð ( kn ) sugklðnei sto R. J deðxoume ìti n. 'Estw ε >. AfoÔ kn, upˆrqei n N ste: gi kˆje n n, (.4.9) kn < ε. AfoÔ h ( n ) eðni koloujð Cuchy, upˆrqei n N ste: gi kˆje n, m n (.4.) n m < ε. Jètoume n = mx{n, n }. 'Estw n n. Tìte k n n n n, ˆr (.4.) kn < ε. EpÐshc k n, n n n, ˆr (.4.) kn n < ε. 'Epeti ìti (.4.3) n n kn + kn < ε + ε = ε. Dhld, n < ε gi kˆje n n. Autì shmðnei ìti n. Je rhm.4.5. Mi koloujð ( n ) sugklðnei n ki mìno n eðni koloujð Cuchy. Apìdeixh. H mð kteôjunsh podeðqthke sthn eisgwg ut c thc prgrˆfou: n upojèsoume ìti n ki n jewr soume tuqìn ε >, upˆrqei n = n (ε) N ste gi kˆje n n n isqôei n < ε. Tìte, gi kˆje n, m n èqoume (.4.4) n m n + m < ε + ε = ε.

UpkoloujÐes ki bsikès koloujðes 'Ar, h ( n ) eðni koloujð Cuchy. Gi thn ntðstrofh kteôjunsh: èstw ( n ) koloujð Cuchy. Apì thn Prìtsh.4.3, h ( n ) eðni frgmènh. Apì to Je rhm Bolzno-Weierstrss, h ( n ) èqei sugklðnous upkoloujð. Tèloc, pì thn Prìtsh.4.4 èpeti ìti h ( n ) sugklðnei. Autì to krit rio sôgklishc eðni polô qr simo. Pollèc forèc jèloume n exsflðsoume thn Ôprxh orðou gi mi koloujð qwrðc n mc endifèrei h tim tou orðou. ArkeÐ n deðxoume ìti h koloujð eðni Cuchy, dhld ìti oi ìroi thc eðni {kontˆ} gi megˆlouc deðktec, kˆti pou den piteð n mntèyoume ek twn protèrwn poiì eðni to ìrio. AntÐjet, gi n doulèyoume me ton orismì tou orðou, prèpei dh n xèroume poiì eðni to upoy fio ìrio (sugkrðnete touc dôo orismoôc: { n } ki {( n ) koloujð Cuchy}.).5 *Prˆrthm: suz thsh gi to xðwm thc plhrìthtc 'Olh mc h douleiˆ xekinˆei me thn {prdoq } ìti to R eðni èn ditetgmèno s m pou iknopoieð to xðwm thc plhrìthtc: kˆje mh kenì, ˆnw frgmèno uposônolì tou èqei elˆqisto ˆnw frˆgm. Qrhsimopoi ntc thn Ôprxh supremum deðxme thn Arqim dei idiìtht: ( ) An R ki ε >, upˆrqei n N ste nε >. Qrhsimopoi ntc ki pˆli to xðwm thc plhrìthtc, deðxme ìti kˆje monìtonh ki frgmènh koloujð sugklðnei. Sn sunèpei p rme to Je rhm Bolzno- Weierstrss: kˆje frgmènh koloujð èqei sugklðnous upkoloujð. Autì me th seirˆ tou mc epètreye n deðxoume thn {idiìtht Cuchy} twn prgmtik n rijm n: ( ) Kˆje koloujð Cuchy prgmtik n rijm n sugklðnei se prgmtikì rijmì. Se ut n thn prˆgrfo j deðxoume ìti to xðwm thc plhrìthtc eðni logik sunèpei twn ( ) ki ( ). An dhld deqtoôme to R sn èn ditetgmèno s m pou èqei thn Arqim dei idiìtht ki thn idiìtht Cuchy, tìte mporoôme n podeðxoume to {xðwm thc plhrìthtc} sn je rhm: Je rhm.5.. 'Estw R èn ditetgmèno s m pou perièqei to Q ki èqei, epiplèon, tic kìloujec idiìthtec:. An R ki ε R, ε >, tìte upˆrqei n N ste nε >.. Kˆje koloujð Cuchy stoiqeðwn tou R sugklðnei se stoiqeðo tou R. Tìte, kˆje mh kenì ki ˆnw frgmèno A R èqei elˆqisto ˆnw frˆgm. Apìdeixh. 'Estw A mh kenì ki ˆnw frgmèno uposônolo tou R. Xekinˆme me tuqìn stoiqeðo A (upˆrqei foô A ). 'Estw b ˆnw frˆgm tou A. Apì thn Sunj kh, upˆrqei k N gi ton opoðo + k > b. Dhld, upˆrqei fusikìc k me thn idiìtht (.5.) gi kˆje A, < + k. Apì thn rq tou elqðstou èpeti ìti upˆrqei elˆqistoc tètoioc fusikìc. Ac ton poôme k. Tìte,

.5 *Prˆrthm: suz thsh gi to xðwm ths plhrìthts Gi kˆje A isqôei < + k. Upˆrqei A ste + (k ). Epgwgikˆ j broôme... n... sto A ki k n N pou iknopoioôn t ex c: Gi kˆje A isqôei < n + n + kn n n. kn. n Apìdeixh tou epgwgikoô b mtoc: 'Eqoume n A ki pì thn Sunj kh upˆrqei elˆqistoc fusikìc k n+ me thn idiìtht: gi kˆje A, (.5.) < n + k n+ n. Autì shmðnei ìti upˆrqei n+ me (.5.3) n + k n+ n n+. Isqurismìc : H ( n ) eðni koloujð Cuchy. Prˆgmti, èqoume (.5.4) n + k n n n < n + k n n, ˆr (.5.5) n n < n. An loipìn n, m N ki n < m, tìte m n m m + m m + + n+ n < m + m + + n < n. An t n, m eðni rketˆ megˆl, utì gðneti ìso jèloume mikrì. Pio sugkekrimèn, n mc d soun ε >, upˆrqei n N t.w / n < ε, opìte gi kˆje n, m n èqoume m n < ε. AfoÔ to R èqei thn idiìtht Cuchy, upˆrqei o = lim n. Isqurismìc : O eðni to elˆqisto ˆnw frˆgm tou A. () O eðni ˆnw frˆgm tou A: c upojèsoume ìti upˆrqei A me >. MporoÔme n broôme ε > ste > + ε. 'Omwc, (.5.6) < n + k n n n + n gi kˆje n N. 'Ar, + ε < n + ( n = + ε lim n + ) n = + ε,

UpkoloujÐes ki bsikès koloujðes to opoðo eðni ˆtopo. (b) An eðni ˆnw frˆgm tou A, tìte : èqoume n gi kˆje n N, ˆr (.5.7) lim n =. Apì t () ki (b) eðni sfèc ìti = sup A..6 Ask seic Omˆd A'. Erwt seic ktnìhshc Exetˆste n oi prkˆtw protˆseic eðni lhjeðc yeudeðc (itiolog ste pl rwc thn pˆnthsh sc).. n + n ki mìno n gi kˆje M > upˆrqoun ˆpeiroi ìroi thc ( n ) pou eðni meglôteroi pì M.. H ( n ) den eðni ˆnw frgmènh n ki mìno n upˆrqei upkoloujð ( kn ) thc ( n ) ste kn +. 3. Kˆje upkoloujð mic sugklðnousc koloujðc sugklðnei. 4. An mi koloujð den èqei fjðnous upkoloujð tìte èqei mi gnhsðwc Ôxous upkoloujð. 5. An h ( n ) eðni frgmènh ki n tìte upˆrqoun b ki upkoloujð ( kn ) thc ( n ) ste kn b. 6. Upˆrqei frgmènh koloujð pou den èqei sugklðnous upkoloujð. 7. An h ( n ) den eðni frgmènh, tìte den èqei frgmènh upkoloujð. 8. 'Estw ( n ) Ôxous koloujð. Kˆje upkoloujð thc ( n ) eðni Ôxous. 9. An h ( n ) eðni Ôxous ki gi kˆpoi upkoloujð ( kn ) thc ( n ) èqoume kn, tìte n.. An n tìte upˆrqei upkoloujð ( kn ) thc ( n ) ste n kn. Omˆd B'. 'Estw ( n ) mi koloujð. DeÐxte ìti n n ki mìno n oi upkoloujðec ( k ) ki ( k ) sugklðnoun sto.. 'Estw ( n ) mi koloujð. Upojètoume ìti oi upkoloujðec ( k ), ( k ) ki ( 3k ) sugklðnoun. DeÐxte ìti: () lim k = lim k = lim 3k. k k k (b) H ( n ) sugklðnei. 3. 'Estw ( n ) mi koloujð. Upojètoume ìti n n+ n+ n gi kˆje n N ki ìti lim ( n n ) =. Tìte h ( n ) sugklðnei se kˆpoion n prgmtikì rijmì pou iknopoieð thn n n gi kˆje n N.

.6 Ask seis 3 4. 'Estw ( n ) mi koloujð ki èstw (x k ) koloujð orik n shmeðwn thc ( n ). Upojètoume oti x k x. DeÐxte oti o x eðni orikì shmeðo thc ( n ). 5. DeÐxte ìti h koloujð ( n ) den sugklðnei ston prgmtikì rijmì, n ki mìno n upˆrqoun ε > ki upkoloujð ( kn ) thc ( n ) ste kn ε gi kˆje n N. 6. 'Estw ( n ) koloujð prgmtik n rijm n ki èstw R. DeÐxte ìti n n ki mìno n kˆje upkoloujð thc ( n ) èqei upkoloujð pou sugklðnei sto. 7. OrÐzoume mi koloujð ( n ) me > ki n+ = + + n. DeÐxte ìti oi upkoloujðec ( k ) ki ( k ) eðni monìtonec ki frgmènec. BreÐte, n upˆrqei, to lim n. n 8. BreÐte to n tero ki to kt tero ìrio twn kolouji n n = ( ) n+ ( + n b n = cos ), ( πn ) + 3 n +, γ n = n (( ) n + ) + n +. n + 9. 'Estw ( n ), (b n ) frgmènec koloujðec. DeÐxte oti lim inf n + lim inf b n lim inf( n + b n ). 'Estw n >, n N. () DeÐxte oti lim sup( n + b n ) lim sup n + lim sup b n. lim inf n+ n lim inf n n lim sup n n lim sup n+ n. (b) An lim n+ n = x, tìte n n x.. 'Estw ( n ) frgmènh koloujð. DeÐxte oti lim sup( n ) = lim inf n ki lim inf( n ) = lim sup n.. 'Estw ( n ) frgmènh koloujð. An deðxte ìti sup X = lim sup n. 3. Qrhsimopoi ntc thn nisìtht X = {x R : x n gi ˆpeirouc n N}, n + + n + + + n,

4 UpkoloujÐes ki bsikès koloujðes deðxte ìti h koloujð n = + + + n den eðni koloujð Cuchy. Sumperˆnte ìti n +. 4. 'Estw < µ < ki koloujð ( n ) gi thn opoð isqôei n+ n µ n n, n. DeÐxte ìti h ( n ) eðni koloujð Cuchy. 5. OrÐzoume =, = b ki n+ = n+n, n. Exetˆste n h ( n ) eðni koloujð Cuchy. Omˆd G' 6. 'Estw m N. BreÐte mi koloujð ( n ) h opoð n èqei krib c m diforetikèc upkoloujðec. 7. 'Estw ( n ) mi koloujð. An sup{ n : n N} = ki n gi kˆje n N, tìte upˆrqei gnhsðwc Ôxous upkoloujð ( kn ) thc ( n ) ste kn. 8. 'Estw ( n ) koloujð jetik n rijm n. JewroÔme to sônolo A = { n : n N}. An inf A =, deðxte ìti h ( n ) èqei fjðnous upkoloujð pou sugklðnei sto. 9. OrÐzoume mi koloujð wc ex c: =, n+ = + n, n = n. BreÐte ìl t orikˆ shmeð thc ( n ). [Upìdeixh: Grˆyte touc dèk pr touc ìrouc thc koloujðc.] 3. 'Estw (x n ) koloujð me thn idiìtht x n+ x n. An < b eðni dôo orikˆ shmeð thc (x n ), deðxte ìti kˆje y [, b] eðni orikì shmeðo thc (x n ). [Upìdeixh: Apgwg se ˆtopo.] 3. () 'Estw A rijm simo uposônolo tou R. DeÐxte ìti upˆrqei koloujð ( n ) ste kˆje x A n eðni orikì shmeðo thc ( n ). (b) DeÐxte ìti upˆrqei koloujð (x n ) ste kˆje x R n eðni orikì shmeðo thc (x n ). 3. 'Estw ( n ) mi koloujð. OrÐzoume b n = sup{ n+k n : k N}. DeÐxte ìti h ( n ) sugklðnei n ki mìno n b n. 33. 'Estw, b >. OrÐzoume koloujð ( n ) me =, = b ki n+ = 4 n+ n, n =,,... 3 Exetˆste n h ( n ) sugklðnei ki n ni, breðte to ìriì thc.

Kefˆlio Seirèc prgmtik n rijm n. SÔgklish seirˆc Orismìc... 'Estw ( k ) mi koloujð prgmtik n rijm n. JewroÔme thn koloujð (..) s n = + + n. Dhld, (..) s =, s = +, s 3 = + + 3,... To sômbolo k eðni h seirˆ me k-ostì ìro ton k. To ˆjroism s n = n k eðni to n-ostì merikì ˆjroism thc seirˆc k ki h (s n ) eðni h koloujð twn merik n jroismˆtwn thc seirˆc k. An h (s n ) sugklðnei se kˆpoion prgmtikì rijmì s, tìte grˆfoume (..3) s = + + + n + s = ki lème ìti h seirˆ sugklðnei (sto s), to de ìrio s = lim s n eðni to ˆjroism thc n seirˆc. An s n + n s n, tìte grˆfoume k = + k = ki lème ìti h seirˆ k poklðnei sto + sto ntðstoiq. An h (s n ) den sugklðnei se prgmtikì rijmì, tìte lème ìti h seirˆ poklðnei. k k Prthr seic... () Pollèc forèc exetˆzoume th sôgklish seir n thc morf c k ìpou m. Se ut n thn perðptwsh jètoume s n+ = + + k k=m

6 Seirès prgmtik n rijm n + n s n m+ = m + m+ + + n (gi n m) ntðstoiq, ki exetˆzoume th sôgklish thc koloujðc (s n ). (b) Apì touc orismoôc pou d sme eðni fnerì ìti gi n exetˆsoume th sôgklish pìklish mic seirˆc, pl c exetˆzoume th sôgklish pìklish mic koloujðc (thc koloujðc (s n ) twn merik n jroismˆtwn thc seirˆc). O n-ostìc ìmwc ìroc thc koloujðc (s n ) eðni èn {ˆjroism me oloèn uxnìmeno m koc}, to opoðo duntoôme (sun jwc) n grˆyoume se kleist morf. Sunep c, h eôresh tou orðou s = lim n s n (ìtn utì upˆrqei) eðni polô suqnˆ nèfikth. Skopìc mc eðni loipìn n nptôxoume kˆpoi krit ri t opoð n mc epitrèpoun (toulˆqiston) n poôme n h (s n ) sugklðnei se prgmtikì rijmì ìqi. Prin proqwr soume se prdeðgmt, j doôme kˆpoiec plèc protˆseic pou j qrhsimopoioôme eleôjer sth sunèqei. An èqoume dôo seirèc k, b k, mporoôme n sqhmtðsoume ton grmmikì sundusmì touc (λ k + µb k ), ìpou λ, µ R. Prìtsh..3. An (..4) Apìdeixh. An s n = k = s ki b k = t, tìte (λ k + µb k ) = λs + µt = λ k + µ b k. n k, t n = n b k ki u n = n (λ k + µb k ) eðni t n-ostˆ merikˆ jroðsmt twn seir n, tìte u n = λs n + µt n. Autì prokôptei eôkol pì tic idiìthtec thc prìsjeshc ki tou pollplsismoô, foô èqoume jroðsmt me pepersmènouc to pl joc ìrouc. 'Omwc, s n s ki t n t, ˆr u n λs + µt. Apì ton orismì tou jroðsmtoc seirˆc èpeti h (..4). Prìtsh..4. () An pleðyoume pepersmèno pl joc {rqik n} ìrwn mic seirˆc, den ephreˆzeti h sôgklish pìklis thc. (b) An llˆxoume pepersmènouc to pl joc ìrouc mic seirˆc, den ephreˆzeti h sôgklish pìklish thc. Apìdeixh. () JewroÔme th seirˆ k. Me th frˆsh {pleðfoume touc rqikoôc ìrouc,,..., m } ennooôme ìti jewroôme thn kinoôrgi seirˆ k=m k. An sumbolðsoume me s n ki t n t n-ostˆ merikˆ jroðsmt twn dôo seir n ntistoðqwc, tìte gi kˆje n m èqoume (..5) s n = + + + m + m + + n = + + m + t n m+. 'Ar h (s n ) sugklðnei n ki mìnon n h (t n m+ ) sugklðnei, dhld n ki mìnon n h (t n ) sugklðnei. EpÐshc, n s n s ki t n t, tìte s = + + + m + t. Dhld, (..6) k = + + m + k. k=m

. SÔgklish seirˆs 7 (b) JewroÔme th seirˆ k. Allˆzoume pepersmènouc to pl joc ìrouc thc ( k ). JewroÔme dhld mi nè seirˆ b k pou ìmwc èqei thn ex c idiìtht: upˆrqei m N ste k = b k gi kˆje k m. An pleðyoume touc pr touc m ìrouc twn dôo seir n, prokôptei h Ðdi seirˆ k. T r, efrmìzoume to (). k=m Prìtsh..5. () An k = s, tìte n. (b) An h seirˆ k sugklðnei, tìte gi kˆje ε > upˆrqei N = N(ε) N ste: gi kˆje n N, (..7) Apìdeixh. () An s n = n k=n+ k < ε. k, tìte s n s ki s n s. 'Ar, (..8) n = s n s n s s =. Sthn prgmtikìtht, utì pou kˆnoume ed eðni n jewr soume mi deôterh koloujð (t n ) h opoð orðzeti wc ex c: dðnoume ujðreth tim ston t gi prˆdeigm, t = ki gi kˆje n jètoume t n = s n. Tìte, t n s (ˆskhsh) ki gi kˆje n èqoume n = s n t n s s = (exhg ste thn pr th isìtht). 'Enc ˆlloc trìpoc gi n podeðxoume ìti n eðni me ton orismì. 'Estw ε >. AfoÔ s n s, upˆrqei n N ste s n s < ε gi kˆje n n. Jètoume n = n +. Tìte, gi kˆje n n èqoume n n ki n n. Sunep c, s s n < ε ki s s n < ε, p ìpou èpeti ìti n = s n s n s n s + s s n < ε + ε = ε gi kˆje n n. Me bˆsh ton orismì, n. (b) An k = s, tìte pì thn (..6) èqoume (..9) β n := k=n+ k = s s n kj c to n. Apì ton orismì tou orðou koloujðc, gi kˆje ε > upˆrqei N = N(ε) N ste: gi kˆje n N, β n < ε. ShmeÐwsh. To mèroc () thc Prìtshc..5 qrhsimopoieðti sn krit rio pìklishc: An h koloujð ( k ) den sugklðnei sto tìte h seirˆ k ngkstikˆ poklðnei.

8 Seirès prgmtik n rijm n PrdeÐgmt () H gewmetrik seirˆ me lìgo x R eðni h seirˆ (..) x k. Dhld k = x k, k =,,,.... An x = tìte s n = n +, en n x èqoume (..) s n = + x + x + + x n = xn+ x. DikrÐnoume dôo peript seic: (i) An x tìte k = x k, dhld k. Apì thn Prìtsh..5() blèpoume ìti h seirˆ (..) poklðnei. (ii) An x < tìte x n+, opìte h (..) deðqnei ìti s n x. Dhld, (..) x k = x. (b) Thleskopikèc seirèc. Upojètoume ìti h koloujð ( k ) iknopoieð thn (..3) k = b k b k+ gi kˆje k N, ìpou (b k ) mi ˆllh koloujð. Tìte, h seirˆ k sugklðnei n ki mìnon n h koloujð (b k ) sugklðnei. Prˆgmti, èqoume (..4) s n = + + n = (b b ) + (b b 3 ) + + (b n b n+ ) = b b n+, opìte b n b n ki mìnon n s n b b. Sn prˆdeigm jewroôme th seirˆ (..5) k = ìpou b k = k. 'Ar,. Tìte, k(k+) k(k + ) = k k + = b k b k+, Dhld, (..6) s n = + + ( n = ) + = n +. ( 3 k(k + ) =. ) ( + + n ) n +

. Seirès me mh rnhtikoôs ìrous 9 Je rhm..6 (krit rio Cuchy). H seirˆ k sugklðnei n ki mìno n isqôei to ex c: gi kˆje ε > upˆrqei N = N(ɛ) N ste: n N m < n tìte n (..7) k = m+ + + n < ε. k=m+ Apìdeixh. An s n = + + + n eðni to n-ostì merikì ˆjroism thc seirˆc, h seirˆ sugklðnei n ki mìnon n h (s n ) sugklðnei. Dhld, n ki mìnon n h (s n ) eðni koloujð Cuchy. Autì ìmwc eðni (pì ton orismì thc koloujðc Cuchy) isodônmo me to ex c: gi kˆje ε > upˆrqei N = N(ɛ) N ste gi kˆje N m < n, (..8) m+ + + n = ( + + n ) ( + + m ) = s n s m < ε. Prˆdeigm: H rmonik seirˆ 'Eqoume k = k. k gi kˆje k N. PrthroÔme ìti: n n > m tìte (..9) m+ + + n = m + + m + + + n n m n. Efrmìzoume to krit rio tou Cuchy. An h rmonik seirˆ sugklðnei, tìte, gi ε = 4, prèpei n upˆrqei N N ste: n N m < n tìte (..) m+ + + n < 4. Epilègoume m = N ki n = N. Tìte, sunduˆzontc tic (..9) ki (..) pðrnoume (..) 4 > N+ + + N N N N =, pou eðni ˆtopo. 'Ar, h rmonik seirˆ poklðnei. ShmeÐwsh: To prˆdeigm thc rmonik c seirˆc deðqnei ìti to ntðstrofo thc Protshc..5() den isqôei. An k den eðni prðtht swstì ìti h seirˆ sugklðnei.. Seirèc me mh rnhtikoôc ìrouc k Se ut thn prˆgrfo suzhtˆme th sôgklish pìklish seir n me mh rnhtikoôc ìrouc. H bsik prt rhsh eðni ìti n gi thn koloujð ( k ) èqoume k gi kˆje k N, tìte h koloujð (s n ) twn merik n jroismˆtwn eðni Ôxous: prˆgmti, gi kˆje n N èqoume (..) s n+ s n = ( + + n + n+ ) ( + + n ) = n+. Je rhm... 'Estw ( k ) koloujð me k gi kˆje k N. H seirˆ k sugklðnei n ki mìnon n h koloujð (s n ) twn merik n jroismˆtwn eðni ˆnw frgmènh. An h (s n ) den eðni ˆnw frgmènh, tìte k = +.

Seirès prgmtik n rijm n Apìdeixh. H (s n ) eðni Ôxous koloujð. An eðni ˆnw frgmènh tìte sugklðnei se prgmtikì rijmì, ˆr h seirˆ sugklðnei. An h (s n ) den eðni ˆnw frgmènh tìte, foô eðni Ôxous, èqoume s n +. ShmeÐwsh. EÐdme ìti mi seirˆ me mh rnhtikoôc ìrouc sugklðnei poklðnei sto +. Epistrèfontc sto prˆdeigm thc rmonik c seirˆc k, blèpoume ìti, foô den sugklðnei, poklðnei sto + : (..) k = +. J d soume mi peujeðc pìdeixh gi to gegonìc ìti h koloujð s n = + + + n teðnei sto +. Pio sugkekrimèn, j deðxoume me epgwg ìti ( ) s n + n gi kˆje n N. Gi n = h nisìtht isqôei wc isìtht: s = +. Upojètoume ìti h ( ) isqôei gi kˆpoion fusikì n. Tìte, s n+ = s n + n + + n + + + n+. Prthr ste ìti o s n+ s n eðni èn ˆjroism n to pl joc rijm n ki ìti o mikrìteroc pì utoôc eðni o. Sunep c, n+ s n+ s n + n n+ = s n + + n + = + n +. 'Ar, h ( ) isqôei gi ton fusikì n +. 'Epeti ìti s n +. AfoÔ h (s n ) eðni Ôxous ki èqei upkoloujð pou teðnei sto +, sumperðnoume ìti s n +..þ Seirèc me fjðnontec mh rnhtikoôc ìrouc Pollèc forèc sunntˆme seirèc k twn opoðwn oi ìroi k fjðnoun proc to : k+ k gi kˆje k N ki k. 'En krit rio sôgklishc pou efrmìzeti suqnˆ se tètoiec peript seic eðni to krit rio sumpôknwshc. Prìtsh.. (Krit rio sumpôknwshc - Cuchy). 'Estw ( k ) mi fjðnous koloujð me k > ki k. H seirˆ k sugklðnei n ki mìno n h seirˆ k k sugklðnei. Apìdeixh. Upojètoume pr t ìti h k k sugklðnei. Tìte, h koloujð twn merik n jroismˆtwn (..3) t n = + + 4 4 + + n n eðni ˆnw frgmènh. 'Estw M èn ˆnw frˆgm thc (t n ). J deðxoume ìti o M eðni ˆnw frˆgm gi t merikˆ jroðsmt thc k. 'Estw s m = + + m. O

. Seirès me mh rnhtikoôs ìrous rijmìc m brðsketi nˆmes se dôo didoqikèc dunˆmeic tou : Ôpˆrqei n N ste n m < n+. Tìte, qrhsimopoi ntc thn upìjesh ìti h ( k ) eðni fjðnous, èqoume s m = + ( + 3 ) + ( 4 + 5 + 6 + 7 ) + + ( n + + n ) +( n + + m ) = + ( + 3 ) + ( 4 + 5 + 6 + 7 ) + + ( n + + n ) M. +( n + + m + + n+ ) + + 4 4 + + n n + n n AfoÔ h k èqei mh rnhtikoôc ìrouc ki h koloujð twn merik n jroismˆtwn thc eðni ˆnw frgmènh, to Je rhm.. deðqnei ìti h k sugklðnei. AntÐstrof: upojètoume ìti h k sugklðnei, dhld ìti h (s m ) eðni ˆnw frgmènh: upˆrqei M R ste s m M gi kˆje m N. Tìte, gi to tuqìn merikì ˆjroism (t n ) thc seirˆc k k èqoume t n = + + 4 4 + + n n + + ( 3 + 4 ) + + ( n + + + n) = s n M. AfoÔ h (t n ) eðni ˆnw frgmènh, to Je rhm.. deðqnei ìti h k k sugklðnei. PrdeÐgmt () k, ìpou p >. 'Eqoume p k = k. AfoÔ p >, h ( p k ) fjðnei proc to. JewroÔme thn (..4) k k = k ( k ) p = ( p )k. H teleutð seirˆ eðni gewmetrik seirˆ me lìgo x p = EÐdme ìti sugklðnei n p x p = <, dhld n p > ki poklðnei n x p p =, dhld n p. p Apì to krit rio sumpôknwshc, h seirˆ sugklðnei n p > ki poklðnei sto + n < p. (b), ìpou k(log k) p >. 'Eqoume p k = k(log k). AfoÔ p >, h ( p k ) fjðnei proc k= to. JewroÔme thn k p (..5) k k = k k (log( k )) p = (log ) p k p.

Seirès prgmtik n rijm n Apì to prohgoômeno prˆdeigm, ut sugklðnei n p > ki poklðnei n p. Apì to krit rio sumpôknwshc, h seirˆ sugklðnei n p > ki poklðnei sto + n < p..bþ O rijmìc e k= k(log k) p 'Eqoume orðsei ton rijmì e wc to ìrio thc gnhsðwc Ôxousc ki ˆnw frgmènhc koloujðc α n := ( + n) n kj c to n. Prìtsh..3. O rijmìc e iknopoieð thn (..6) e = Apìdeixh. JumhjeÐte ìti! =. Grˆfoume s n gi to n-ostì merikì ˆjroism thc seirˆc sto dexiì mèloc: k!. (..7) s n = +! +! + + n!. Apì to diwnumikì nˆptugm, èqoume ( + n) n = + dhld, ( ) n n + ( ) n n + + ( ) n n n n = + n n(n ) n(n ) (n k + ) + + +! n! n k! n k n(n ) + + n! n n = +! + ( ) + + [( ) ( n )]! n n! n n +! +! + + n!, (..8) α n s n. 'Estw n N. O prohgoômenoc upologismìc deðqnei ìti n k > n tìte ( + ) k = + k! + ( ) + + [(! k n! k + + [( ) ( k )] k! k k +! + ( ) + + [(! k n! k Krt ntc to n stjerì ki f nontc to k, blèpoume ìti ( (..9) e = lim + ) k + k k! +! + + n! = s n. ) ( n )] k ) ( n )]. k

. Seirès me mh rnhtikoôs ìrous 3 AfoÔ h Ôxous koloujð (s n ) eðni ˆnw frgmènh pì ton e, èpeti ìti h (s n ) sugklðnei ki lim s n e. Apì thn ˆllh pleurˆ, h (..8) deðqnei ìti e = lim α n n n lim s n. 'Ar, n (..) e = lim n s n = ìpwc isqurðzeti h Prìtsh. Qrhsimopoi ntc ut n thn nprˆstsh tou e, j deðxoume ìti eðni ˆrrhtoc rijmìc. Prìtsh..4. O e eðni ˆrrhtoc. k!, Apìdeixh. Upojètoume ìti o e eðni rhtìc. Tìte, upˆrqoun m, n N ste (..) e = m n = Dhld, (..) ( m n = +! + + ) ( + n! k!. (n + )! + + (n + s)! + Pollplsiˆzontc t dôo mèlh thc (..) me n!, mporoôme n grˆyoume < A = [ ( m n! n +! + + )] n! = n + + (n + )(n + ) + + (n + ) (n + s) +. Prthr ste ìti, pì ton trìpo orismoô tou, o [ ( m (..3) A = n! n +! + + )] n! eðni fusikìc rijmìc. 'Omwc, gi kˆje s N èqoume n + + (n + )(n + ) + + (n + ) (n + s) 'Ar, (..4) ). + 6 + 3 + + s < 3 + 8 k = 3 + 4 =. n + + (n + )(n + ) + + (n + ) (n + s) +. 'Epeti ìti o fusikìc rijmìc A iknopoieð thn (..5) < A ki èqoume ktl xei se ˆtopo.

4 Seirès prgmtik n rijm n.3 Genikˆ krit ri.3þ Apìluth sôgklish seirˆc Orismìc.3.. Lème ìti h seirˆ sugklðnei. Lème ìti h seirˆ sugklðnei polôtwc. k sugklðnei polôtwc n h seirˆ k k sugklðnei upì sunj kh n sugklðnei llˆ den H epìmenh prìtsh deðqnei ìti h pìluth sôgklish eðni isqurìterh pì thn (pl ) sôgklish. Prìtsh.3.. An h seirˆ k sugklðnei polôtwc, tìte h seirˆ k sugklðnei. Apìdeixh. J deðxoume ìti iknopoieðti to krit rio Cuchy (Je rhm..6). 'Estw ε >. AfoÔ h seirˆ k sugklðnei, upˆrqei N N ste: gi kˆje N m < n, (.3.) n k=m+ k < ε. Tìte, gi kˆje N m < n èqoume n n (.3.) k k < ε. k=m+ k=m+ 'Ar h seirˆ k iknopoieð to krit rio Cuchy. Apì to Je rhm..6, sugklðnei. PrdeÐgmt () H seirˆ èqoume (.3.3) ( ) k k sugklðnei. MporoÔme n elègxoume ìti sugklðnei polôtwc: ( ) k k = k ki h teleutð seirˆ sugklðnei (eðni thc morf c (b) H seirˆ (.3.4) ( ) k k den sugklðnei polôtwc, foô ( ) k k = k k me p = > ). p

.3 Genikˆ krit ri 5 (rmonik seirˆ). MporoÔme ìmwc n deðxoume ìti h seirˆ sugklðnei upì sunj kh. JewroÔme pr t to merikì ˆjroism 'Epeti ìti s m = m ( ) k (.3.5) s m+ = s m + k = + 3 4 + + m m = + 3 4 + 5 6 + + (m )m. (m + )(m + ) > s m, dhld, h upkoloujð (s m ) eðni gnhsðwc Ôxous. (s m ) eðni ˆnw frgmènh, foô (.3.6) s m < + 3 + 5 + + (m ), PrthroÔme epðshc ìti h ki to dexiì mèloc thc (.3.6) frˆsseti pì to (m )-ostì merikì ˆjroism thc seirˆc h opoð sugklðnei. 'Ar h upkoloujð (s m ) sugklðnei se kˆpoion k prgmtikì rijmì s. Tìte, (.3.7) s m = s m + m s + = s. AfoÔ oi upkoloujðec (s m ) ki (s m ) twn ˆrtiwn ki twn peritt n ìrwn thc (s m ) sugklðnoun ston s, sumperðnoume ìti s n s..3bþ Krit ri sôgkrishc Je rhm.3.3 (krit rio sôgkrishc). JewroÔme tic seirèc ìpou b k > gi kˆje k N. Upojètoume ìti upˆrqei M > ste k ki b k, (.3.8) k M b k gi kˆje k N ki ìti h seirˆ b k sugklðnei. Tìte, h seirˆ k sugklðnei polôtwc. Apìdeixh. Jètoume s n = n k ki t n = n b k. Apì thn (.3.8) èpeti ìti (.3.9) s n M t n gi kˆje n N. AfoÔ h seirˆ b k sugklðnei, h koloujð (t n ) eðni ˆnw frgmènh. Apì thn (.3.9) sumperðnoume ìti ki h (s n ) eðni ˆnw frgmènh. 'Ar, h k sugklðnei.

6 Seirès prgmtik n rijm n Je rhm.3.4 (orikì krit rio sôgkrishc). JewroÔme tic seirèc b k, ìpou b k > gi kˆje k N. Upojètoume ìti k ki k (.3.) lim = l R k b k ki ìti h seirˆ b k sugklðnei. Tìte, h seirˆ k sugklðnei polôtwc. Apìdeixh. H koloujð M > ste (.3.) ( ) k b k sugklðnei, ˆr eðni frgmènh. k b k M Dhld, upˆrqei gi kˆje k N. Tìte, iknopoieðti h (.3.8) ki mporoôme n efrmìsoume to Je rhm.3.3. Je rhm.3.5 (isodônmh sumperiforˆ). JewroÔme tic seirèc k ki b k, ìpou k, b k > gi kˆje k N. Upojètoume ìti k (.3.) lim = l >. k b k Tìte, h seirˆ b k sugklðnei n ki mìno n h seirˆ k sugklðnei. Apìdeixh. An h b k sugklðnei, tìte h k sugklðnei pì to Je rhm.3.4. AntÐstrof, c upojèsoume ìti h k sugklðnei. AfoÔ k b k l >, èqoume b k k. Enllˆssontc touc rìlouc twn l ( k) ki (b k ), blèpoume ìti h b k sugklðnei, qrhsimopoi ntc xnˆ to Je rhm.3.4. PrdeÐgmt () Exetˆzoume th sôgklish thc seirˆc (.3.) AfoÔ h sin(kx) k k sin(kx) k k. sin(kx) k, ìpou x R. PrthroÔme ìti sugklðnei, sumperðnoume (pì to krit rio sôgkrishc) ìti h seirˆ sugklðnei polôtwc. (b) Exetˆzoume th sôgklish thc seirˆc k+ ki k 4 +k +3 b k = k, tìte 3 k+ k 4 +k +3. PrthroÔme ìti n k = (.3.3) k b k = k4 + k 3 k 4 + k + 3.

.3 Genikˆ krit ri 7 AfoÔ h k 3 k+ sugklðnei. k 4 +k +3 sugklðnei, sumperðnoume (pì to orikì krit rio sôgkrishc) ìti h (g) Tèloc, exetˆzoume th sôgklish thc seirˆc k+ k +. 'Opwc sto prohgoômeno prˆdeigm, n jewr soume tic koloujðec b k = k+ ki k + k =, tìte k (.3.4) Apì to Je rhm.3.5 èpeti ìti h dhld poklðnei..3gþ k = k + b k k + k >. Krit rio lìgou ki krit rio rðzc k+ èqei thn Ðdi sumperiforˆ me thn k + Je rhm.3.6 (Krit rio lìgou - D Alembert). 'Estw k mi seirˆ me mh mhdenikoôc ìrouc. () An lim <, tìte h k sugklðnei polôtwc. (b) An lim k+ k k k+ k k >, tìte h k poklðnei. Apìdeixh. () Upojètoume ìti lim k+ k k = l <. 'Estw x > me l < x <. Tìte, upˆrqei N N ste: k+ k x giˆ kˆje k N. Dhld, (.3.5) N+ x N, N+ x N+ x N klp. Epgwgikˆ deðqnoume ìti (.3.6) k x k N N = N x N giˆ kˆje k N. SugkrÐnoume tic seirèc k=n k ki k=n (.3.7) k M x k gi kˆje k N, ìpou M = N x N. H seirˆ xk x k. Apì thn (.3.6) blèpoume ìti k=n, k x k sugklðnei, diìti proèrqeti pì thn gewmetrik seirˆ x k (me ploif twn pr twn ìrwn thc) ki < x <. 'Ar, h k sugklðnei. 'Epeti ìti h k sugklðnei ki ut. k=n (b) AfoÔ lim k+ k >, upˆrqei N N ste k+ k gi kˆje k N. Dhld, k (.3.8) k k N >

8 Seirès prgmtik n rijm n gi kˆje k N. Tìte, k ki, pì thn Prìtsh..5(), h k poklðnei. ShmeÐwsh. An lim k+ k k thc k. Prthr ste ìti h sugklðnei ki k+ k = k Prˆdeigm =, prèpei n exetˆsoume lli c th sôgklish pìklish (k+). Exetˆzoume th sôgklish thc seirˆc (.3.9) 'Ar, h seirˆ sugklðnei. k poklðnei ki k+ k = k k+, en h. 'Eqoume k! k+ k = k! (k + )! = k + <. Prt rhsh. H pìdeixh tou Jewr mtoc.3.6, qwrðc ousistik mettrop, dðnei to ex c isqurìtero potèlesm: 'Estw k mi seirˆ me mhdenikoôc ìrouc. () An lim sup k+ k <, tìte h seirˆ k sugklðnei polôtwc. Prˆgmti, n k jewr soume x > me l < x <, tìte pì ton qrkthrismì tou lim sup, upˆrqei N N ste x gi kˆje k N. SuneqÐzoume thn pìdeixh ìpwc prin. k+ k (b) An lim inf k+ k k >, tìte h seirˆ k poklðnei. Prˆgmti, n jewr soume x > me l > x >, tìte pì ton qrkthrismì tou lim inf, upˆrqei N N ste k+ k x > gi kˆje k N. SuneqÐzoume thn pìdeixh ìpwc prin. Je rhm.3.7 (krit rio rðzc - Cuchy). 'Estw k mi seirˆ prgmtik n rijm n. () An lim k k (b) An lim k k k <, tìte h seirˆ sugklðnei polôtwc. k >, tìte h seirˆ poklðnei. k Apìdeixh () Epilègoume x > me thn idiìtht lim k N N ste k k x gi kˆje k N. IsodÔnm, (.3.) k x k gi kˆje k n. SugkrÐnoume tic seirèc k=n k ki k k < x <. Tìte, upˆrqei k=n x k. AfoÔ x <, h deôterh seirˆ sugklðnei. 'Ar h k sugklðnei. 'Epeti ìti h k sugklðnei polôtwc. k=n k (b) AfoÔ lim k >, upˆrqei N N ste k k gi kˆje k N. k Dhld, k telikˆ. 'Ar k ki h k poklðnei.

.3 Genikˆ krit ri 9 k ShmeÐwsh. An lim k =, prèpei n exetˆsoume lli c th sôgklish pìklish k thc k. Gi tic, k k èqoume k k. H pr th poklðnei en h deôterh sugklðnei. PrdeÐgmt x () Exetˆzoume th sôgklish thc seirˆc k, ìpou k x R. 'Eqoume k k = x k k k x. An x <, tìte lim k = x < ki h seirˆ sugklðnei polôtwc. k k An x >, tote lim k = x > ki h seirˆ poklðnei. An x =, to k krit rio rðzc den dðnei sumpèrsm. Gi x = pðrnoume thn rmonik seirˆ h opoð poklðnei. Gi x = pðrnoume thn {enllˆssous seirˆ} opoð sugklðnei. 'Ar, h seirˆ sugklðnei n ki mìno n x <. ( ) k k x (b) Exetˆzoume th sôgklish thc seirˆc k k, ìpou x R. 'Eqoume k k = x k x k. 'Ar, lim k = x. An x > h seirˆ poklðnei. An x < h k k seirˆ sugklðnei polôtwc. An x = to krit rio rðzc den dðnei sumpèrsm. Sthn perðptwsh x = ± h seirˆ pðrnei th morf sugklðnei polôtwc ìtn x. k k, dhld sugklðnei. 'Ar, h seirˆ Prt rhsh. H pìdeixh tou Jewr mtoc.3.7, qwrðc ousistik mettrop, dðnei to ex c isqurìtero potèlesm: 'Estw k mi seirˆ me mhdenikoôc ìrouc. k () An lim sup k <, tìte h seirˆ k sugklðnei polôtwc. Prˆgmti, n k jewr soume x > me l < x <, tìte pì ton qrkthrismì tou lim sup, upˆrqei N N ste k x k gi kˆje k N. SuneqÐzoume thn pìdeixh ìpwc prin. k >, tìte h seirˆ k (b) An lim sup k poklðnei. Prˆgmti, n jewr soume k x > me l > x >, tìte pì ton qrkthrismì tou lim sup, upˆrqoun ˆpeiroi deðktec k < k < < k n < k n+ < ste kn x kn > gi kˆje n N. 'Ar, n ki efrmìzeti to krit rio pìklishc..3dþ To krit rio tou Dirichlet To krit rio tou Dirichlet exsflðzei (merikèc forèc) th sôgklish mic seirˆc h opoð den sugklðnei polôtwc (sugklðnei upì sunj kh). L mm.3.8 (ˆjroish ktˆ mèrh - Abel). 'Estw ( k ) ki (b k ) dôo koloujðec. OrÐzoume s n = + + n, s =. Gi kˆje m < n, isqôei h isìtht h (.3.) n k b k = k=m n k=m s k (b k b k+ ) + s n b n s m b m.

3 Seirès prgmtik n rijm n Apìdeixh. Grˆfoume n k b k = k=m = = = pou eðni to zhtoômeno. n (s k s k )b k k=m n s k b k k=m n s k b k k=m n k=m n s k b k k=m n k=m s k b k+ s k (b k b k+ ) + s n b n s m b m, Je rhm.3.9 (krit rio Dirichlet). 'Estw ( k ) ki (b k ) dôo koloujðec me tic ex c idiìthtec: () H (b k ) èqei jetikoôc ìrouc ki fjðnei proc to. (b) H koloujð twn merik n jroismˆtwn s n = + + n thc ( k ) eðni frgmènh: upˆrqei M > ste (.3.3) s n M gi kˆje n N. Tìte, h seirˆ k b k sugklðnei. Apìdeixh. J qrhsimopoi soume to krit rio tou Cuchy. 'Estw ε >. Qrhsimopoi ntc thn upìjesh (), brðskoume N N ste (.3.4) 'An N m < n, tìte n k b k = k=m ε M > b N b N+ b N+ >. n s k (b k b k+ ) + s n b n s m b m k=m n k=m s k b k b k+ + s n b n + s m b m n M (b k b k+ ) + Mb n + Mb m k=m = Mb m < M ε M = ε. Apì to krit rio tou Cuchy, h seirˆ k b k sugklðnei.

.3 Genikˆ krit ri 3 Prˆdeigm (krit rio Leibniz) Seirèc me enllssìmen prìshm ( ) k b k, ìpou h {b k } fjðnei proc to. T merikˆ jroðsmt thc (( ) k ) eðni frgmèn, foô s n = n o n eðni ˆrtioc ki s n = n o n eðni perittìc. 'Ar, kˆje tètoi seirˆ sugklðnei. Prˆdeigm, h seirˆ.3eþ ( ) k k. *Dekdik prˆstsh prgmtik n rijm n Skopìc mc se ut n thn prˆgrfo eðni n deðxoume ìti kˆje prgmtikìc rijmìc èqei dekdik prˆstsh: eðni dhld ˆjroism seirˆc thc morf c (.3.5) k k = + + +, ìpou Z ki k {,,..., 9} gi kˆje k. Prthr ste ìti kˆje seirˆ ut c thc morf c sugklðnei ki orðzei ènn prgmtikì rijmì x = k. Prˆgmti, h gewmetrik seirˆ sugklðnei ki k k epeid k 9 gi kˆje k, h seirˆ k sugklðnei sômfwn me to k k k krit rio sôgkrishc seir n. L mm.3.. An N ki k {,,..., 9} gi kˆje k N, tìte (.3.6) k=n k k N. H rister nisìtht isqôei sn isìtht n ki mìnon n k = gi kˆje k N, en h dexiˆ nisìtht isqôei sn isìtht n ki mìnon n k = 9 gi kˆje k N. Apìdeixh. 'Eqoume (.3.7) k=n k k k=n k =. An k = gi kˆje k N, tìte m N, tìte k=n k=n k =. AntÐstrof, n k m gi kˆpoion k k = m m + = k=n k m m + k=n k m m >. k k k

3 Seirès prgmtik n rijm n Apì thn ˆllh pleurˆ, (.3.8) k=n k k k=n An k = 9 gi kˆje k N, tìte (.3.9) k=n 9 k = 9 ( N + + ) + = N. k k = k=n AntÐstrof, n m 8 gi kˆpoion m N, tìte 9 k = N. k=n k k = m m + k=n k m 8 m + k=n k m = m + k=n k k = m + < N, 9 k = 9 m m + 9 k N ki utì sumplhr nei thn pìdeixh tou L mmtoc. k=n k m 9 k L mm.3.. 'Estw n mh rnhtikìc kèrioc ki èstw N. Tìte upˆrqoun kèrioi p, p,... p n ste: p k {,,..., 9} gi k N, p N ki (.3.3) n = N p N + N p N + + p + p. Apìdeixh. Diir ntc didoqikˆ me pðrnoume n = q + p, ìpou p 9 ki q q = q + p, ìpou p 9 ki q q = q 3 + p, ìpou p 9 ki q 3.. q N = p N + p N, ìpou p N 9 ki q N. Epgwgikˆ, èqoume: n = q + p = q + p + p = 3 q 3 + p + p + p = = N q N + N p N + p + p. Jètontc p N = q N èqoume to zhtoômeno. Qrhsimopoi ntc t L mmt.3. ki.3. j deðxoume ìti kˆje prgmtikìc rijmìc èqei dekdik prˆstsh.

.3 Genikˆ krit ri 33 Je rhm.3.. () Kˆje prgmtikìc rijmìc x grˆfeti sn ˆjroism {dekdik c seirˆc}: (.3.3) x = k k = + + +, ìpou N {} ki k {,,..., 9} gi kˆje k. Tìte, lème ìti o x èqei th dekdik prˆstsh x =. 3. (b) Oi rijmoð thc morf c x = m ìpou m N ki N èqoun krib c dôo N dekdikèc prstˆseic: (.3.3) x =. N 9999 =. N ( N + ) 'Oloi oi ˆlloi mh rnhtikoð prgmtikoð rijmoð èqoun mondik dekdik prˆstsh. Apìdeixh. () 'Estw x. Upˆrqei mh rnhtikìc kèrioc, to kèrio mèroc tou x, ste: (.3.33) x < +. QwrÐzoume to diˆsthm [, + ) se Ðs upodist mt m kouc. O x n kei se èn pì utˆ. 'Ar, upˆrqei {,,..., 9} ste (.3.34) + x < + +. QwrÐzoume to nèo utì diˆsthm (pou èqei m koc ) se Ðs upodist mt m kouc. O x n kei se èn pì utˆ, ˆr upˆrqei {,,..., 9} ste (.3.35) + + x < + + +. SuneqÐzontc epgwgikˆ, gi kˆje k brðskoume k {,,..., 9} ste (.3.36) + + + k k x < + + + k + k. Apì thn ktskeu, t merikˆ jroðsmt s n thc seirˆc k h opoð dhmiourgeð- k ti, iknopoioôn thn s n x < s n + n. 'Ar, (.3.37) x s n < n. 'Epeti ìti s n x, dhld (.3.38) x = k k. (b) Ac upojèsoume ìti kˆpoioc x èqei toulˆqiston dôo diforetikèc dekdikèc prstˆseic. Dhld, (.3.39) x =. = b.b b,

34 Seirès prgmtik n rijm n ìpou, b N {}, k, b k {,,..., 9} gi kˆje k, ki upˆrqei m me thn idiìtht m b m. 'Estw N o elˆqistoc m gi ton opoðo m b m. Dhld, (.3.4) = b, = b,..., N = b N, N b N. QwrÐc periorismì thc genikìthtc upojètoume ìti N < b N. Apì thn (.3.4) k=n ki pì to L mm.3. èpeti ìti k k = k=n b k k N b N N N k = k = k=n+ N N. 'Ar, ìlec oi nisìthtec eðni isìthtec. Dhld, (.3.4) b N N = k=n+ b k k ki (.3.43) k=n+ k k = N, k=n+ b k k =. Apì to L mm.3., b N = N +, k = 9, n k N +, b k =, n k N +. 'Ar, n o x èqei perissìterec pì mð dekdikèc prstˆseic, tìte èqei krib c dôo prstˆseic, tic kìloujec: (.3.44) x =. N 999 =. N ( N + ) Tìte, o x isoôti me x = + + + + N N + N + N gi kˆpoiouc m N ki N. = N + N + + N + N + N m = N

.4 Dunmoseirès 35 AntÐstrof, èstw ìti x = mporoôme n grˆyoume m, ìpou m N ki N. Apì to L mm.3. N (.3.45) m = N p N + N p N + + p + p, ìpou p N N {} ki p k {,,..., 9} gi k N. An p m eðni o pr toc mh mhdenikìc ìroc thc koloujðc p, p,..., p N, p N, tìte x = N p N + + m p m N = p N + p N + + p m N m = p N.p N p m = p N.p N (p m ) 99. Autì oloklhr nei thn pìdeixh tou (b)..4 Dunmoseirèc Orismìc.4.. 'Estw ( k ) mi koloujð prgmtik n rijm n. H seirˆ (.4.) k x k lègeti dunmoseirˆ me suntelestèc k. O x eðni mi prˆmetroc pì to R. To prìblhm pou j suzht soume ed eðni: gi dojeðs koloujð suntelest n ( k ) n brejoôn oi timèc tou x gi tic opoðec h ntðstoiqh dunmoseirˆ sugklðnei. Gi kˆje tètoio x lème ìti h dunmoseirˆ sugklðnei sto x. Prìtsh.4.. 'Estw k x k mi dunmoseirˆ me suntelestèc k. () An h dunmoseirˆ sugklðnei sto y ki n x < y, tìte h dunmoseirˆ sugklðnei polôtwc sto x. (b) An h dunmoseirˆ poklðnei sto y ki n x > y, tìte h dunmoseirˆ poklðnei sto x. Apìdeixh. () AfoÔ h k y k sugklðnei, èqoume k y k. 'Ar, upˆrqei N N ste (.4.) k y k gi kˆje k N. 'Estw x R me x < y. Gi kˆje k N èqoume (.4.3) k x k = k y k x y H gewmetrik seirˆ èpeti to sumpèrsm. x y k=n k sugklðnei, diìti k x y x y k. <. Apì to krit rio sôgkrishc

36 Seirès prgmtik n rijm n (b) An h dunmoseirˆ sunèkline sto x, pì to () j sunèkline polôtwc sto y, ˆtopo. 'Estw k x k mi dunmoseirˆ me suntelestèc k. Me bˆsh thn Prìtsh.4. mporoôme n deðxoume ìti to sônolo twn shmeðwn st opoð sugklðnei h dunmoseirˆ eðni {ousistikˆ} èn diˆsthm summetrikì wc proc to (, endeqomènwc, to {} to R). Autì fðneti wc ex c: orðzoume (.4.4) R := sup{ x : h dunmoseirˆ sugklðnei sto x}. To sônolo sto dexiì mèloc eðni mh kenì, foô h dunmoseirˆ sugklðnei sto. H Prìtsh.4. deðqnei ìti n x < R tìte h dunmoseirˆ sugklðnei polôtwc sto x. Prˆgmti, pì ton orismì tou R upˆrqei y me R y > x ste h dunmoseirˆ n sugklðnei sto y, opìte efrmìzeti h Prìtsh.4.() sto x. Apì ton orismì tou R eðni fnerì ìti n x > R tìte h dunmoseirˆ poklðnei sto x. 'Ar, h dunmoseirˆ sugklðnei se kˆje x ( R, R) ki poklðnei se kˆje x me x > R. To diˆsthm ( R, R) onomˆzeti diˆsthm sôgklishc thc dunmoseirˆc. H suz thsh pou kˆnme deðqnei ìti to sônolo sôgklishc thc dunmoseirˆc, dhld to sônolo ìlwn twn shmeðwn st opoð sugklðnei, prokôptei pì to ( R, R) me thn prosj kh (Ðswc) tou R tou R twn ±R. Sthn perðptwsh pou R = +, h dunmoseirˆ sugklðnei se kˆje x R. Sthn perðptwsh pou R =, h dunmoseirˆ sugklðnei mìno sto shmeðo x =. To prìblhm eðni loipìn t r to ex c: p c mporoôme n prosdiorðsoume thn ktðn sôgklishc mic dunmoseirˆc sunrt sei twn suntelest n thc. Mi pˆnthsh mc dðnei to krit rio thc rðzc gi th sôgklish seir n. Je rhm.4.3. 'Estw k x k mi dunmoseirˆ me suntelestèc k. Upojètoume k ìti upˆrqei to lim k = ki jètoume R = me th sômbsh ìti = + ki k + =. () An x ( R, R) h dunmoseirˆ sugklðnei polôtwc sto x. (b) An x / [ R, R] h dunmoseirˆ poklðnei sto x. Apìdeixh. Efrmìzoume to krit rio thc rðzc gi th sôgklish seir n. Exetˆzoume mìno thn perðptwsh < < + (oi peript seic = ki = + f nonti sn ˆskhsh). () An x < R tìte k (.4.5) lim k x k k = x lim k = x = x k k R <. Apì to krit rio thc rðzc, h k x k sugklðnei polôtwc. (b) An x > R tìte k (.4.6) lim k x k = x k R >. Apì to krit rio thc rðzc, h k x k poklðnei.

.5 Ask seis 37 Prt rhsh.4.4. To Je rhm.4.3 den mc epitrèpei n sumperˆnoume mèswc tic sumbðnei st {ˆkr ±R tou dist mtoc sôgklishc}. 'Opwc deðqnoun t epìmen prdeðgmt, mporeð h dunmoseirˆ n sugklðnei se èn, se knèn ki st dôo ˆkr.. Gi thn x k elègqoume ìti R =. Gi x = ± èqoume tic seirèc k ki ( ) k oi opoðec poklðnoun.. Gi thn elègqoume ìti R =. Gi x = ± èqoume tic seirèc x k (k+) (k + ) ki ( ) k (k + ) oi opoðec sugklðnoun. 3. Gi thn elègqoume ìti R =. Gi x = ± èqoume tic seirèc x k k+ k + ki H pr th poklðnei, en h deôterh sugklðnei. ( ) k k +. AntÐstoiqo potèlesm prokôptei n qrhsimopoi soume to krit rio tou lìgou sth jèsh tou krithrðou thc rðzc. Je rhm.4.5. 'Estw k x k mi dunmoseirˆ me suntelestèc k. Upojètoume ìti upˆrqei to lim k+ k = ki jètoume R =. k () An x ( R, R) h dunmoseirˆ sugklðnei polôtwc sto x. (b) An x / [ R, R] h dunmoseirˆ poklðnei sto x. Apìdeixh. Efrmìste to krit rio tou lìgou gi th sôgklish seir n..5 Ask seic A' Omˆd. Erwt seic ktnìhshc 'Estw ( k ) mi koloujð prgmtik n rijm n. Exetˆste n oi prkˆtw protˆseic eðni lhjeðc yeudeðc (itiolog ste pl rwc thn pˆnthsh sc).. An k tìte h koloujð s n = + + n eðni frgmènh.. An h koloujð s n = + + n eðni frgmènh tìte h seirˆ k sugklðnei.

38 Seirès prgmtik n rijm n 3. An k, tìte h seirˆ k sugklðnei polôtwc. 4. An h seirˆ k sugklðnei, tìte h seirˆ k sugklðnei. 5. An k > gi kˆje k N ki n < k+ k k sugklðnei. < gi kˆje k N, tìte h seirˆ 6. An k > gi kˆje k N ki n lim k+ k k =, tìte h seirˆ k poklðnei. 7. An k > gi kˆje k N ki n k+ k +, tìte h h seirˆ k poklðnei. 8. An k, tìte h seirˆ ( ) k k sugklðnei. 9. An k > gi kˆje k N ki n h seirˆ k sugklðnei, tìte h seirˆ sugklðnei.. An h seirˆ k sugklðnei, tìte h seirˆ k sugklðnei.. An h seirˆ k sugklðnei ki n ( kn ) eðni mi upkoloujð thc ( n ), tìte h seirˆ kn sugklðnei.. An k > gi kˆje k N ki n h seirˆ k sugklðnei, tìte h seirˆ sugklðnei. 3. H seirˆ 4 6 (k) k! sugklðnei. 4. H seirˆ k( + k ) p sugklðnei n ki mìno n p <. k k B' Omˆd 5. DeÐxte ìti n lim b k = b tìte (b k b k+ ) = b b. k 6. DeÐxte ìti () (k )(k+) = (b) k +3 k 6 k = 3 (g) k+ k k +k =.

.5 Ask seis 39 7. UpologÐste to ˆjroism thc seirˆc. k(k+)(k+) 8. Exetˆste gi poièc timèc tou prgmtikoô rijmoô x sugklðnei h seirˆ 9. Efrmìste t krit ri lìgou ki rðzc stic kìloujec seirèc: () k k x k (b) x k k! (g) x k k (d) k 3 x k (e) k k! xk (st) k x k k (z) k 3 x k (h) 3 k k x k k!.. +x k An gi kˆpoiec timèc tou x R knèn pì utˆ t dôo krit ri den dðnei pˆnthsh, exetˆste th sôgklish pìklish thc seirˆc me ˆllo trìpo.. Exetˆste n sugklðnoun poklðnoun oi seirèc ki + 3 + + 3 + 3 + 3 3 + 4 + 3 4 + + + 8 + 4 + 3 + 6 + 8 + 64 +.. N brejeð ikn ki ngkð sunj kh gi thn koloujð ( n ) ste n sugklðnei h seirˆ + + 3 3 +.. Exetˆste n sugklðnei poklðnei h seirˆ k stic prkˆtw peript seic: () k = k + k n= (b) k = + k k (g) k = k+ k k (d) k = ( k k ) k. 3. Exetˆste n sugklðnoun poklðnoun oi seirèc k + k k 3, ( k k ), cos k k, k! k k. 4. Exetˆste wc proc th sôgklish tic prkˆtw seirèc. 'Opou emfnðzonti oi prˆmetroi p, q, x R n brejoôn oi timèc touc gi tic opoðec oi ntðstoiqec seirèc sugklðnoun. () (d) (z) ( ) + k k (e) k + k ) k k+ k p ( (b) p k k p ( < p) (g) k= ( < q < p) (st) p k q k k p k q ( < q < p) +( ) k k (h) ( k k p + k + k ).

4 Seirès prgmtik n rijm n 5. 'Estw ìti k gi kˆje k N. DeÐxte ìti h seirˆ k +k k sugklðnei. 6. OrÐzoume mi koloujð ( k ) wc ex c: n o k eðni tetrˆgwno fusikoô rijmoô jètoume k = ki n o k k den eðni tetrˆgwno fusikoô rijmoô jètoume k = k. Exetˆste n sugklðnei h seirˆ k. 7. Exetˆste n sugklðnei poklðnei h seirˆ ( ) k k p, ìpou p R. 8. 'Estw { k } fjðnous koloujð pou sugklðnei sto. OrÐzoume s = DeÐxte ìti ( ) n (s s n ) n+. ( ) k k. 9. 'Estw ( k ) fjðnous koloujð jetik n rijm n. DeÐxte ìti: n h sugklðnei tìte k k. 3. 'Estw ìti k > gi kˆje k N. An h k sugklðnei, deðxte ìti oi k sugklðnoun epðshc. k, k + k, k + k 3. Upojètoume ìti k gi kˆje k N ki ìti h seirˆ k sugklðnei. DeÐxte ìti h seirˆ k k+ sugklðnei. DeÐxte ìti, n h { k } eðni fjðnous, tìte isqôei ki to ntðstrofo. 3. Upojètoume ìti k gi kˆje k N ki ìti h seirˆ k sugklðnei. DeÐxte ìti h seirˆ k k sugklðnei. 33. Upojètoume ìti k gi kˆje k N ki ìti h seirˆ k poklðnei. DeÐxte ìti k ( + )( + ) ( + k ) =.

.5 Ask seis 4 G' Omˆd 34. 'Estw ( k ) fjðnous koloujð jetik n rijm n me k. DeÐxte ìti: n h k poklðnei tìte { min k, } = +. k 35. Upojètoume ìti k > gi kˆje k N ki ìti h k poklðnei. Jètoume s n = + + + n. () DeÐxte ìti h k + k poklðnei. (b) DeÐxte ìti: gi m < n, ki sumperˆnte ìti h (g) DeÐxte ìti n s n s n s n m+ s m+ + + n s n s m s n k s k poklðnei. ki sumperˆnte ìti h k s k sugklðnei. 36. Upojètoume ìti k > gi kˆje k N ki ìti h k sugklðnei. Jètoume r n = k. k=n () DeÐxte ìti: gi m < n, ki sumperˆnte ìti h m r m + + n r n r n+ r m k r k poklðnei. (b) DeÐxte ìti rn n < ( rn ) r n+ ki sumperˆnte ìti h k rk sugklðnei. 37. 'Estw ( k ) koloujð prgmtik n rijm n. DeÐxte ìti n h seirˆ poklðnei tìte ki h seirˆ k k poklðnei. k 38. 'Estw ( k ) koloujð jetik n prgmtik n rijm n. DeÐxte ìti n h seirˆ k sugklðnei, tìte ki h sugklðnei. k k+ k

4 Seirès prgmtik n rijm n 39. 'Estw ( k ) h koloujð pou orðzeti pì tic k = k ki k = k. Exetˆste n h seirˆ ( ) k k sugklðnei. 4. Upojètoume ìti k gi kˆje k N. OrÐzoume b k = k k m=k+ DeÐxte ìti h k sugklðnei n ki mìno n h b k sugklðnei. [Upìdeixh: An s n ki t n eðni t merikˆ jroðsmt twn dôo seir n, dokimˆste n sugkrðnete t s n ki t n.] 4. 'Estw ( k ) koloujð jetik n prgmtik n rijm n. JewroÔme thn koloujð m. b k = + + + k k. k(k + ) DeÐxte ìti: n h k sugklðnei, tìte h seirˆ b k sugklðnei ki t jroðsmt twn dôo seir n eðni Ðs. 4. 'Estw ( k ) koloujð jetik n rijm n ste k = + ki k. DeÐxte ìti n α < β tìte upˆrqoun fusikoð m n ste α < n k < β. k=m 43. DeÐxte ìti n α < β tìte upˆrqoun fusikoð m n ste α < m + m + + + n < β.

Kefˆlio 3 Omoiìmorfh sunèqei 3. Omoiìmorfh sunèqei Prin d soume ton orismì thc omoiìmorfhc sunèqeic, j exetˆsoume pio prosektikˆ dôo plˆ prdeðgmt suneq n sunrt sewn. () JewroÔme th sunˆrthsh f(x) = x, x R. GnwrÐzoume ìti h f eðni suneq c sto R, kˆti pou eôkol epibebi noume usthrˆ qrhsimopoi ntc ton orismì thc sunèqeic: 'Estw x R ki èstw ε >. Zhtˆme δ > ste (3..) x x < δ = f(x) f(x ) < ε, dhld x x < ε. H epilog tou δ eðni profn c: rkeð n pˆroume δ = ε. Prthr ste ìti to δ pou br kme exrtˆti mìno pì to ε pou dìjhke ki ìqi pì to sugkekrimèno shmeðo x. H sunˆrthsh f metbˆlleti me ton {Ðdio rujmì} se olìklhro to pedðo orismoô thc: n x, y R ki x y < ε, tìte f(x) f(y) < ε. (b) JewroÔme t r th sunˆrthsh g(x) = x, x R. EÐni pˆli gnwstì ìti h g eðni suneq c sto R (foô g = f f). An jel soume n to epibebi soume me ton eyilontikì orismì, jewroôme x R ki ε >, ki zhtˆme δ > me thn idiìtht (3..) x x < δ = x x < ε. 'Enc trìpoc gi n epilèxoume ktˆllhlo δ eðni o ex c. SumfwnoÔme pì thn rq ìti j pˆroume < δ, opìte An loipìn epilèxoume x x = x x x + x ( x + x ) x x ( x + ) x x. (3..3) δ = min tìte { } ε,, x + (3..4) x x < δ = x x < ( x + )δ ε.

44 Omoiìmorfh sunèqei 'Ar, h g eðni suneq c sto x. Prthr ste ìmwc ìti to δ pou epilèxme den exrtˆti mìno pì to ε pou mc dìjhke, llˆ ki pì to shmeðo x sto opoðo elègqoume thn sunèqei thc g. H epilog pou kˆnme sthn (3..3) deðqnei ìti ìso pio mkriˆ brðsketi to x pì to, tìso pio mikrì prèpei n epilèxoume to δ. J mporoôse bèbi n pei kneðc ìti Ðswc upˆrqei klôteroc trìpoc epilog c tou δ, kìm ki nexˆrthtoc pì to shmeðo x. Ac doôme to Ðdio prìblhm me ènn deôtero trìpo. JewroÔme x > ki ε >. MporoÔme n upojèsoume ìti ε < x, foô t mikrˆ ε eðni utˆ pou prousiˆzoun endifèron. MporoÔme epðshc n koitˆme mìno x >, foô mc endifèrei ti gðneti kontˆ sto x to opoðo èqei upotejeð jetikì. H nisìtht x x < ε iknopoieðti n ki mìno n x ε < x < x + ε, dhld n ki mìno n (3..5) x ε < x < x + ε. IsodÔnm, n ( ) (3..6) x x ε < x x < x + ε x. Autì sumbðnei n ki mìno n x x < min { x { = min } x ε, x + ε x ε x ε + x, } ε x + ε + x = ε x + ε + x. Upojèsme ìti x > ε. 'Ar, (3..7) ε x < < x = x x + ε + x x. + ε + x x An loipìn x x < ε, tìte x x x < x x > ki o prohgoômenoc +ε+x upologismìc deðqnei ìti x x < ε. Dhld, n < ε < x tìte h klôterh epilog tou δ sto shmeðo x eðni (3..8) δ = δ(ε, x ) = ε x + ε + x. Den mporoôme n exsflðsoume thn (3..) n epilèxoume meglôtero δ. An t prohgoômen dôo epiqeir mt den eðni polôtwc peistikˆ, dðnoume ki èn trðto. Isqurismìc. JewroÔme thn g(x) = x, x R. 'Estw ε >. Den upˆrqei δ > me thn idiìtht: n x, y R ki y x < δ tìte g(y) g(x) < ε. Prthr ste ìti o isqurismìc eðni isodônmoc me to ex c: gi dojèn ε > den upˆrqei kˆpoi omoiìmorfh epilog tou δ pou n mc epitrèpei n elègqoume thn (3..) se kˆje x R.

3. Omoiìmorfh sunèqei 45 Apìdeixh tou isqurismoô. Ac upojèsoume ìti upˆrqei δ > ste: n x, y R ki y x < δ tìte g(y) g(x) < ε. AfoÔ gi kˆje x R èqoume x + δ x = δ < δ, prèpei, gi kˆje x R n isqôei h ( (3..9) x + δ x ) < ε. Eidikìter, gi kˆje x > prèpei n isqôei h ( (3..) δx < δx + δ 4 = x + δ x ) < ε. 'Omwc tìte, gi kˆje x > j eðqme (3..) x < ε δ. Autì eðni ˆtopo: to R j tn ˆnw frgmèno. T prdeðgmt pou d sme deðqnoun mi {prˆleiy } mc ston orismì thc sunèqeic. 'Enc pio prosektikìc orismìc j tn o ex c: H f : A R eðni suneq c sto x A n gi kˆje ε > upˆrqei δ(ε, x ) > ste: n x A ki x x < δ, tìte f(x) f(x ) < ε. O sumbolismìc δ(ε, x ) j èdeiqne ìti to δ exrtˆti tìso pì to ε ìso ki pì to shmeðo x. Oi sunrt seic (ìpwc h f(x) = x) pou mc epitrèpoun n epilègoume to δ nexˆrtht pì to x lègonti omoiìmorf suneqeðc: Orismìc 3... 'Estw f : A R mi sunˆrthsh. Lème ìti h f eðni omoiìmorf suneq c sto A n gi kˆje ε > mporoôme n broôme δ = δ(ε) > ste (3..) n x, y A ki x y < δ tìte f(x) f(y) < ε. PrdeÐgmt () H f(x) = x eðni omoiìmorf suneq c sto R. (b) H g(x) = x den eðni omoiìmorf suneq c sto R. (g) JewroÔme th sunˆrthsh g(x) = x tou (b), periorismènh ìmwc sto kleistì diˆsthm [ M, M], ìpou M >. Tìte, gi kˆje x, y [ M, M] èqoume (3..3) g(y) g(x) = y x = x + y y x M y x. DÐneti ε >. An epilèxoume δ(ε) = ε [ M, M] ki x y < δ èqoume M tìte h (3..3) deðqnei ìti n x, y (3..4) g(y) g(x) M y x < Mδ = ε. Dhld, h g eðni omoiìmorf suneq c sto [ M, M]. To prˆdeigm (g) odhgeð ston ex c orismì. Orismìc 3... 'Estw f : A R mi sunˆrthsh. Lème ìti h f eðni Lipschitz suneq c n upˆrqei M > ste: gi kˆje x, y A (3..5) f(x) f(y) M x y.

46 Omoiìmorfh sunèqei Prìtsh 3..3. Kˆje Lipschitz suneq c sunˆrthsh eðni omoiìmorf suneq c. Apìdeixh. 'Estw f : A R ki M > ste f(x) f(y) M x y gi kˆje x, y A. An mc d soun ε >, epilègoume δ = ε M. Tìte, gi kˆje x, y A me x y < δ èqoume (3..6) f(x) f(y) M x y < Mδ = ε. 'Epeti ìti h f eðni omoiìmorf suneq c sto A. H epìmenh Prìtsh mc dðnei èn qr simo krit rio gi n exsflðzoume ìti mi sunˆrthsh eðni Lipschitz suneq c (ˆr, omoiìmorf suneq c). Prìtsh 3..4. 'Estw I èn diˆsthm ki èstw f : I R prgwgðsimh sunˆrthsh. Upojètoume ìti h f eðni frgmènh: upˆrqei stjerˆ M > ste: f (x) M gi kˆje eswterikì shmeðo x tou I. Tìte, h f eðni Lipschitz suneq c me stjerˆ M. Apìdeixh. 'Estw x < y sto I. Apì to je rhm mèshc tim c upˆrqei ξ (x, y) ste (3..7) f(y) f(x) = f (ξ)(y x). Tìte, (3..8) f(y) f(x) = f (ξ) y x M y x. SÔmfwn me ton Orismì 3.., h f eðni Lipschitz suneq c me stjerˆ M. Apì th suz thsh pou prohg jhke tou orismoô thc omoiìmorfhc sunèqeic, eðni logikì n perimènoume ìti oi omoiìmorf suneqeðc sunrt seic eðni suneqeðc. Autì podeiknôeti me pl sôgkrish twn dôo orism n: Prìtsh 3..5. An h f : A R eðni omoiìmorf suneq c, tìte eðni suneq c. Apìdeixh. Prˆgmti: èstw x A ki ε >. Apì ton orismì thc omoiìmorfhc sunèqeic, upˆrqei δ > ste n x, y A ki x y < δ tìte f(x) f(y) < ε. Epilègoume utì to δ. An x A ki x x < δ, tìte f(x) f(x ) < ε (pˆrte y = x ). AfoÔ to ε > tn tuqìn, h f eðni suneq c sto x. 3. Qrkthrismìc thc omoiìmorfhc sunèqeic mèsw kolouji n JumhjeÐte ton qrkthrismì thc sunèqeic mèsw kolouji n: n f : A R, tìte h f eðni suneq c sto x A n ki mìno n gi kˆje koloujð (x n ) me x n A ki x n x, isqôei f(x n ) f(x ). O ntðstoiqoc qrkthrismìc thc omoiìmorfhc sunèqeic èqei wc ex c: Je rhm 3... 'Estw f : A R mi sunˆrthsh. H f eðni omoiìmorf suneq c sto A n ki mìno n gi kˆje zeugˆri kolouji n (x n ), (y n ) sto A me x n y n isqôei (3..) f(x n ) f(y n ).

3. Qrkthrismìs ths omoiìmorfhs sunèqeis mèsw kolouji n 47 Apìdeixh. Upojètoume pr t ìti h f eðni omoiìmorf suneq c sto A. 'Estw (x n ), (y n ) dôo koloujðec sto A me x n y n. J deðxoume ìti f(x n ) f(y n ) : 'Estw ε >. Apì ton orismì thc omoiìmorfhc sunèqeic, upˆrqei δ > ste (3..) n x, y A ki x y < δ tìte f(x) f(y) < ε. AfoÔ x n y n, upˆrqei n (δ) N ste: n n n tìte x n y n < δ. 'Estw n n. Tìte, x n y n < δ ki x n, y n A, opìte h (3..) dðnei (3..3) f(x n ) f(y n ) < ε. AfoÔ to ε > tn tuqìn, sumperðnoume ìti f(x n ) f(y n ). AntÐstrof: c upojèsoume ìti (3..4) n x n, y n A ki x n y n tìte f(x n ) f(y n ). J deðxoume ìti h f eðni omoiìmorf suneq c sto A. 'Estw ìti den eðni. Tìte, upˆrqei ε > me thn ex c idiìtht: Gi kˆje δ > upˆrqoun x δ, y δ A me x δ y δ < δ llˆ f(x δ ) f(y δ ) ε. Epilègontc didoqikˆ δ =,,..., n,..., brðskoume zeugˆri x n, y n A ste (3..5) x n y n < n llˆ f(x n) f(y n ) ε. JewroÔme tic koloujðec (x n ), (y n ). Apì thn ktskeu èqoume x n y n, llˆ pì thn f(x n ) f(y n ) ε gi kˆje n N blèpoume ìti den mporeð n isqôei h f(x n ) f(y n ) (exhg ste gitð). Autì eðni ˆtopo, ˆr h f eðni omoiìmorf suneq c sto A. PrdeÐgmt () JewroÔme th sunˆrthsh f(x) = x sto (, ]. H f eðni suneq c llˆ den eðni omoiìmorf suneq c. Gi n to doôme, rkeð n broôme dôo koloujðec (x n ), (y n ) sto (, ] pou n iknopoioôn thn x n y n llˆ n mhn iknopoioôn thn x n y n. PÐrnoume x n = ki n y n =. Tìte, n x n, y n (, ] ki (3..6) x n y n = n n = n llˆ (3..7) f(x n ) f(y n ) = x n y n = n n = n. (b) JewroÔme th sunˆrthsh g(x) = x sto R. OrÐzoume x n = n + n ki y n = n. Tìte, (3..8) x n y n = n llˆ (3..9) g(x n ) g(y n ) = ( n + ) n = + n n.

48 Omoiìmorfh sunèqei 'Ar, h g den eðni omoiìmorf suneq c sto R. (g) OrÐzoume f(x) = cos(x ), x R. H f eðni suneq c sto R ki f(x) gi kˆje x R. Dhld, h f eðni epiplèon frgmènh. 'Omwc h f den eðni omoiìmorf suneq c: gi n to deðte, jewr ste tic koloujðec (3..) x n = (n + )π ki y n = nπ. Tìte, (3..) x n y n = (n + )π nπ = llˆ (n + )π nπ π =, (n + )π + nπ (n + )π + nπ (3..) f(x n ) f(y n ) = cos((n + )π) cos(nπ) = gi kˆje n N. Apì to Je rhm 3.. èpeti to sumpèrsm. Upˆrqoun loipìn frgmènec suneqeðc sunrt seic pou den eðni omoiìmorf suneqeðc (sqediˆste th grfik prˆstsh thc cos(x ) gi n deðte to lìgo: gi megˆl x, h f nebðnei pì thn tim sthn tim ki ktebðnei pì thn tim sthn tim ìlo ki pio gr gor - o rujmìc metbol c thc gðneti polô megˆloc). 3.3 SuneqeÐc sunrt seic se kleistˆ dist mt Sthn prˆgrfo 3. eðdme ìti h sunˆrthsh g(x) = x den eðni omoiìmorf suneq c sto I = R llˆ eðni omoiìmorf suneq c se kˆje diˆsthm thc morf c I = [ M, M], M > (osod pote megˆlo ki n eðni to M). Autì pou isqôei genikˆ eðni ìti kˆje suneq c sunˆrthsh f : [, b] R eðni omoiìmorf suneq c: Je rhm 3.3.. 'Estw f : [, b] R suneq c sunˆrthsh. Tìte, h f eðni omoiìmorf suneq c sto [, b]. Apìdeixh. Ac upojèsoume ìti h f den eðni omoiìmorf suneq c. Tìte, ìpwc sthn pìdeixh tou Jewr mtoc 3.., mporoôme n broôme ε > ki dôo koloujðec (x n ), (y n ) sto [, b] me x n y n ki f(x n ) f(y n ) ε gi kˆje n N. AfoÔ x n, y n b gi kˆje n N, oi (x n ) ki (y n ) eðni frgmènec koloujðec. Apì to Je rhm Bolzno-Weierstrss, upˆrqei upkoloujð (x kn ) thc (x n ) h opoð sugklðnei se kˆpoio x R. AfoÔ x kn b gi kˆje n, sumperðnoume ìti x b. Dhld, (3.3.) x kn x [, b]. Prthr ste ìti x kn y kn, ˆr (3.3.) y kn = x kn (x kn y kn ) x = x. Apì th sunèqei thc f sto x èpeti ìti (3.3.3) f(x kn ) f(x) ki f(y kn ) f(x). Dhld, (3.3.4) f(x kn ) f(y kn ) x x =.

3.3 SuneqeÐs sunrt seis se kleistˆ dist mt 49 Autì eðni ˆtopo, foô f(x kn ) f(y kn ) ε gi kˆje n N. 'Ar, h f eðni omoiìmorf suneq c sto [, b]. Prt rhsh. To gegonìc ìti h f tn orismènh sto kleistì diˆsthm [, b] qrhsimopoi jhke me dôo trìpouc. Pr ton, mporèsme n broôme sugklðnousec upkoloujðec twn (x n ), (y n ) (je rhm Bolzno-Weierstrss). DeÔteron, mporoôsme n poôme ìti to koinì ìrio x ut n twn upkolouji n exkoloujeð n brðsketi sto pedðo orismoô [, b] thc f. Qrhsimopoi sme dhld to ex c: (3.3.5) n z n b ki z n z, tìte z b. To epìmeno je rhm podeiknôei ìti oi omoiìmorf suneqeðc sunrt seic èqoun thn ex c {kl idiìtht}: peikonðzoun koloujðec Cuchy se koloujðec Cuchy. Autì den isqôei gi ìlec tic suneqeðc sunrt seic: jewr ste thn f(x) = x sto (, ]. H x n = eðni koloujð n Cuchy sto (, ], ìmwc h f(x n) = n den eðni koloujð Cuchy. Je rhm 3.3.. 'Estw f : A R omoiìmorf suneq c sunˆrthsh ki èstw (x n ) koloujð Cuchy sto A. Tìte, h (f(x n )) eðni koloujð Cuchy. Apìdeixh. 'Estw ε >. Upˆrqei δ > ste: n x, y A ki x y < δ tìte f(x) f(y) < ε. H (x n ) eðni koloujð Cuchy, ˆr upˆrqei n (δ) ste (3.3.6) n m, n n (δ), tìte x n x m < δ. 'Omwc tìte, (3.3.7) f(x n ) f(x m ) < ε. Br kme n N me thn idiìtht (3.3.8) n m, n n (δ) tìte f(x n ) f(x m ) < ε. AfoÔ to ε > tn tuqìn, h (f(x n )) eðni koloujð Cuchy. EÐdme ìti kˆje suneq c sunˆrthsh f orismènh se kleistì diˆsthm eðni omoiìmorf suneq c. J exetˆsoume to ex c er thm: 'Estw f : (, b) R suneq c sunˆrthsh. P c mporoôme n elègxoume n h f eðni omoiìmorf suneq c sto (, b)? Je rhm 3.3.3. 'Estw f : (, b) R suneq c sunˆrthsh. H f eðni omoiìmorf suneq c sto (, b) n ki mìno n upˆrqoun t lim f(x) ki lim f(x). x + x b Apìdeixh. Upojètoume pr t ìti upˆrqoun t lim f(x) ki lim f(x). OrÐzoume x + x b mi {epèktsh} g thc f sto [, b], jètontc: g() = lim f(x), g(b) = lim f(x) x + x b ki g(x) = f(x) n x (, b). H g eðni suneq c sto kleistì diˆsthm [, b] (exhg ste gitð), ˆr omoiìmorf suneq c. J deðxoume ìti h f eðni ki ut omoiìmorf suneq c sto (, b). 'Estw ε >. AfoÔ h g eðni omoiìmorf suneq c, upˆrqei δ > ste: n x, y [, b] ki x y < δ tìte g(x) g(y) < ε. JewroÔme x, y (, b) me x y < δ. Tìte, pì ton orismì thc g èqoume (3.3.9) f(x) f(y) = g(x) g(y) < ε.

5 Omoiìmorfh sunèqei AntÐstrof, upojètoume ìti h f eðni omoiìmorf suneq c sto (, b) ki deðqnoume ìti upˆrqei to lim x + f(x) (h Ôprxh tou ˆllou pleurikoô orðou podeiknôeti me ton Ðdio trìpo). J deðxoume ìti n (x n ) eðni koloujð sto (, b) me x n, tìte h (f(x n )) sugklðnei. Autì eðni ˆmeso pì to Je rhm 3.3.: h (x n ) sugklðnei, ˆr h (x n ) eðni koloujð Cuchy, ˆr h (f(x n )) eðni koloujð Cuchy, ˆr h (f(x n )) sugklðnei se kˆpoion prgmtikì rijmì l. EpÐshc, to ìrio thc (f(x n )) eðni nexˆrthto pì thn epilog thc (x n ): èstw (y n ) mi ˆllh koloujð sto (, b) me y n. Tìte, x n y n. Apì to Je rhm 3.., (3.3.) f(x n ) f(y n ). Xèroume dh ìti lim n f(x n) = l, ˆr (3.3.) f(y n ) = f(x n ) (f(x n ) f(y n )) l + = l. Apì thn rq thc metforˆc (gi to ìrio sunˆrthshc) èpeti ìti lim f(x) = l. x + PrdeÐgmt () JewroÔme th sunˆrthsh f(x) = x sto [, ]. H f eðni suneq c sto [, ], epomènwc eðni omoiìmorf suneq c. 'Omwc, h f den eðni Lipschitz suneq c sto [, ]. An tn, j up rqe M > ste (3.3.) f(x) f(y) M x y gi kˆje x, y [, ]. Eidikìter, gi kˆje n N j eðqme ( ) (3.3.3) f n f() = n = n n M n. Dhld, n M gi kˆje n N. Autì eðni ˆtopo: to N j tn ˆnw frgmèno. (b) H sunˆrthsh f(x) = x eðni Lipschitz suneq c sto [, + ), ˆr omoiìmorf suneq c. Prˆgmti, n x tìte (3.3.4) f (x) = x, dhld h f èqei frgmènh prˆgwgo sto [, + ). Apì thn Prìtsh 3..4 eðni Lipschitz suneq c me stjerˆ /. (g) Ac doôme t r thn Ðdi sunˆrthsh f(x) = x sto [, + ). H f den eðni Lipschitz suneq c sto [, + ) oôte mporoôme n efrmìsoume to Je rhm 3.3.. EÐdme ìmwc ìti h f eðni omoiìmorf suneq c sto [, ] ki omoiìmorf suneq c sto [, + ). Autì ftˆnei gi n deðxoume ìti eðni omoiìmorf suneq c sto [, + ): 'Estw ε >. Upˆrqei δ > ste: n x, y [, ] ki x y < δ tìte f(x) f(y) < ε (pì thn omoiìmorfh sunèqei thc f sto [, ]). EpÐshc, upˆrqei δ > ste: n x, y [, + ) ki x y < δ tìte f(x) f(y) < ε (pì thn omoiìmorfh sunèqei thc f sto [, + )). Jètoume δ = min{δ, δ } >. 'Estw x < y [, + ) me x y < δ. DikrÐnoume treðc peript seic:

3.4 Sustolès je rhm stjeroô shmeðou 5 (i) An x < y ki x y < δ, tìte x y < δ ˆr f(x) f(y) < ε < ε. (ii) An x < y ki x y < δ, tìte x y < δ ˆr f(x) f(y) < ε < ε. (iii) An x < < y ki x y < δ, prthroôme ìti x < δ ki y < δ. 'Omwc, x, [, ] ki, y [, + ). MporoÔme loipìn n grˆyoume f(x) f(y) f(x) f() + f() f(y) < ε + ε = ε. 3.4 Sustolèc je rhm stjeroô shmeðou Orismìc 3.4.. Mi sunˆrthsh f : A R lègeti sustol n upˆrqei < M < ste: gi kˆje x, y A (3.4.) f(x) f(y) M x y. Profn c, kˆje sustol eðni Lipschitz suneq c. Je rhm 3.4. (je rhm stjeroô shmeðou). 'Estw f : R R sustol. Upˆrqei mondikì y R me thn idiìtht (3.4.) f(y) = y. Apìdeixh. Apì thn upìjesh upˆrqei < M < ste f(x) f(y) M x y gi kˆje x, y R. H f eðni Lipschitz suneq c, ˆr omoiìmorf suneq c. Epilègoume tuqìn x R. OrÐzoume mi koloujð (x n ) mèsw thc (3.4.3) x n+ = f(x n ), n N. Tìte, (3.4.4) x n+ x n = f(x n ) f(x n ) M x n x n gi kˆje n. Epgwgikˆ podeiknôoume ìti (3.4.5) x n+ x n M n x x gi kˆje n. 'Epeti ìti n n > m sto N, tìte x n x m x n x n + + x m+ x m (M n + + M m ) x x = M n m M M m x x M m M x x. AfoÔ < M <, èqoume M m. 'Ar, gi dojèn ε > mporoôme n broôme n (ε) ste: n n > m n tìte M m M x x < ε, ki sunep c, x n x m < ε. Epomènwc, h (x n ) eðni koloujð Cuchy ki utì shmðnei ìti sugklðnei: upˆrqei y R ste x n y. J deðxoume ìti f(y) = y: pì thn x n y ki th sunèqei thc f sto y blèpoume ìti f(x n ) f(y). 'Omwc x n+ = f(x n ) ki x n+ y, ˆr f(x n ) y. Apì th mondikìtht tou orðou koloujðc prokôptei h f(y) = y. To y eðni to mondikì stjerì shmeðo thc f. 'Estw z y me f(z) = z. Tìte, (3.4.6) < z y = f(z) f(y) M z y, dhld M, to opoðo eðni ˆtopo.

5 Omoiìmorfh sunèqei 3.5 Ask seic Omˆd A'. DeÐxte to je rhm mègisthc ki elˆqisthc tim c gi mi suneq sunˆrthsh f : [, b] R qrhsimopoi ntc to je rhm Bolzno Weiertstrss: () DeÐxte pr t ìti upˆrqei M > ste f(x) M gi kˆje x [, b], me pgwg se ˆtopo. An utì den isqôei, mporoôme n broôme x n [, b] ste f(x n ) > n, n =,,.... H (x n ) èqei upkoloujð (x kn ) ste x kn x [, b]. Qrhsimopoi - ste thn rq thc metforˆc (h f eðni suneq c sto x ) gi n ktl xete se ˆtopo. (b) Apì to () èqoume M := sup{f(x) : x [, b]} <. Tìte, mporoôme n broôme x n [, b] ste f(x n ) M (exhg ste gitð). H (x n ) èqei upkoloujð (x kn ) ste x kn x [, b]. Qrhsimopoi ste thn rq thc metforˆc (h f eðni suneq c sto x ) gi n sumperˆnete ìti f(x ) = M. Autì podeiknôei ìti h f pðrnei mègisth tim (sto x ). (g) Ergzìmenoi ìmoi, deðxte ìti h f pðrnei elˆqisth tim.. 'Estw X R. Lème ìti mi sunˆrthsh f : X R iknopoieð sunj kh Lipschitz n upˆrqei M ste: gi kˆje x, y X, f(x) f(y) M x y. DeÐxte ìti n h f : X R iknopoieð sunj kh Lipschitz tìte eðni omoiìmorf suneq c. IsqÔei to ntðstrofo? 3. 'Estw f : [, b] R suneq c, prgwgðsimh sto (, b). DeÐxte ìti h f iknopoieð sunj kh Lipschitz n ki mìno n h f eðni frgmènh. 4. 'Estw n N, n ki f(x) = x /n, x [, ]. DeÐxte ìti h sunˆrthsh f den iknopoieð sunj kh Lipschitz. EÐni omoiìmorf suneq c? 5. Exetˆste n oi prkˆtw sunrt seic iknopoioôn sunj kh Lipschitz: () f : [, ] R me f(x) = x sin x n x ki f() =. (b) g : [, ] R me g(x) = x sin x n x ki g() =. 6. 'Estw f : [, b] [m, M] ki g : [m, M] R omoiìmorf suneqeðc sunrt seic. DeÐxte ìti h g f eðni omoiìmorf suneq c. 7. 'Estw f, g : I R omoiìmorf suneqeðc sunrt seic. DeÐxte ìti () h f + g eðni omoiìmorf suneq c sto I. (b) h f g den eðni ngkstikˆ omoiìmorf suneq c sto I, n ìmwc oi f, g upotejoôn ki frgmènec tìte h f g eðni omoiìmorf suneq c sto I. 8. 'Estw f : R R suneq c sunˆrthsh me thn ex c idiìtht: gi kˆje ε > upˆrqei M = M(ε) > ste n x M tìte f(x) < ε. DeÐxte ìti h f eðni omoiìmorf suneq c. 9. 'Estw R ki f : [, + ) R suneq c sunˆrthsh me thn ex c idiìtht: upˆrqei to lim f(x) ki eðni prgmtikìc rijmìc. DeÐxte ìti h f eðni omoiìmorf x + suneq c.

3.5 Ask seis 53. 'Estw f : R R omoiìmorf suneq c sunˆrthsh. DeÐxte ìti upˆrqoun A, B > ste f(x) A x + B gi kˆje x R.. 'Estw n N, n >. Qrhsimopoi ntc thn prohgoômenh 'Askhsh deðxte ìti h sunˆrthsh f(x) = x n, x R den eðni omoiìmorf suneq c.. () 'Estw f : [, + ) R suneq c sunˆrthsh. Upojètoume ìti upˆrqei > ste h f n eðni omoiìmorf suneq c sto [, + ). DeÐxte ìti h f eðni omoiìmorf suneq c sto [, + ). (b) DeÐxte ìti h f(x) = x eðni omoiìmorf suneq c sto [, + ). 3. 'Estw f : (, b) R omoiìmorf suneq c sunˆrthsh. DeÐxte ìti upˆrqei suneq c sunˆrthsh ˆf : [, b] R ste ˆf(x) = f(x) gi kˆje x (, b). 4. Exetˆste n oi prkˆtw sunrt seic eðni omoiìmorf suneqeðc. (i) f : R R me f(x) = 3x +. (ii) f : [, + ) R me f(x) = x. (iii) f : (, π] R me f(x) = x sin x. (iv) f : (, ) R me f(x) = sin x. (v) f : (, ) R me f(x) = x sin x. (vi) f : (, ) R me f(x) = sin x x. (vii) f : (, ) R me f(x) = cos(x3 ) x. (viii) f : R R me f(x) = x +4. (ix) f : R R me f(x) = x. + x (x) f : [, ] R me f(x) = (xi) f : R R me f(x) = x sin x. x. x + (xii) f : [, + ) R me f(x) = cos(x ) x+. Omˆd B'. Erwt seic ktnìhshc Exetˆste n oi prkˆtw protˆseic eðni lhjeðc yeudeðc (itiolog ste pl rwc thn pˆnths sc). 5. H sunˆrthsh f(x) = x + x eðni omoiìmorf suneq c sto (, ). 6. H sunˆrthsh f(x) = x eðni omoiìmorf suneq c sto (, ). 7. An h sunˆrthsh f den eðni frgmènh sto (, ), tìte h f den eðni omoiìmorf suneq c sto (, ). 8. An h (x n ) eðni koloujð Cuchy ki h f eðni omoiìmorf suneq c sto R, tìte h (f(x n )) eðni koloujð Cuchy.

54 Omoiìmorfh sunèqei 9. An h f eðni omoiìmorf suneq c sto (, ), tìte to lim n f( n) upˆrqei.. JewroÔme tic f(x) = x ki g(x) = sin x. Oi f ki g eðni omoiìmorf suneqeðc sto R, ìmwc h fg den eðni omoiìmorf suneq c sto R.. H sunˆrthsh f : R R me f(x) = x n x > ki f(x) = x n x, eðni omoiìmorf suneq c sto R.. Kˆje frgmènh ki suneq c sunˆrthsh f : R R eðni omoiìmorf suneq c. Omˆd G' 3. DeÐxte ìti h sunˆrthsh f : (, ) (, ) R me f(x) = n x (, ) ki f(x) = n x (, ) eðni suneq c llˆ den eðni omoiìmorf suneq c. 4. 'Estw f : [, b] R suneq c sunˆrthsh ki èstw ε >. DeÐxte ìti mporoôme n qwrðsoume to [, b] se pepersmèn to pl joc didoqikˆ upodist mt tou idðou m kouc ètsi ste: n t x, y n koun sto Ðdio upodiˆsthm, tìte f(x) f(y) < ε. 5. 'Estw f : R R suneq c, frgmènh ki monìtonh sunˆrthsh. DeÐxte ìti h f eðni omoiìmorf suneq c. 6. 'Estw f : R R suneq c ki periodik sunˆrthsh. Dhld, upˆrqei T > ste f(x + T ) = f(x) gi kˆje x R. DeÐxte ìti h f eðni omoiìmorf suneq c. 7. 'Estw X R frgmèno sônolo ki f : X R omoiìmorf suneq c sunˆrthsh. DeÐxte ìti h f eðni frgmènh: upˆrqei M > ste f(x) M gi kˆje x X. 8. 'Estw A mh kenì uposônolo tou R. OrÐzoume f : R R me f(x) = inf{ x : A} (f(x) eðni h {pìstsh} tou x pì to A). DeÐxte ìti () f(x) f(y) x y gi kˆje x, y R. (b) h f eðni omoiìmorf suneq c.

Kefˆlio 4 Olokl rwm Riemnn 4. O orismìc tou Drboux Se ut n thn prˆgrfo dðnoume ton orismì tou oloklhr mtoc Riemnn gi frgmènec sunrt seic pou orðzonti se èn kleistì diˆsthm. Gi mi frgmènh sunˆrthsh f : [, b] R me mh rnhtikèc timèc, j jèlme to olokl rwm n dðnei to embdìn tou qwrðou pou perikleðeti nˆmes sto grˆfhm thc sunˆrthshc, ton orizìntio ˆxon y = ki tic ktkìrufec eujeðec x = ki x = b. Orismìc 4... () 'Estw [, b] èn kleistì diˆsthm. Dimèrish tou [, b] j lème kˆje pepersmèno uposônolo (4..) P = {x, x,..., x n } tou [, b] me x = ki x n = b. J upojètoume pˆnt ìti t x k P eðni ditetgmèn wc ex c: (4..) = x < x < < x k < x k+ < < x n = b. J grˆfoume (4..3) P = { = x < x < < x n = b} gi n tonðsoume ut n krib c th diˆtxh. Prthr ste ìti pì ton orismì, kˆje dimèrish P tou [, b] perièqei toulˆqiston dôo shmeð: to ki to b (t ˆkr tou [, b]). (b) Kˆje dimèrish P = { = x < x < < x n = b} qwrðzei to [, b] se n upodist mt [x k, x k+ ], k =,,..., n. Onomˆzoume plˆtoc thc dimèrishc P to meglôtero pì t m kh ut n twn upodisthmˆtwn. Dhld, to plˆtoc thc dimèrishc isoôti me (4..4) P := mx{x x, x x,..., x n x n }. Prthr ste ìti den pitoôme n ispèqoun t x k (t n upodist mt den èqoun prðtht to Ðdio m koc). (g) H dimèrish P lègeti eklèptunsh thc P n P P, dhld n h P prokôptei pì thn P me thn prosj kh kˆpoiwn (pepersmènwn to pl joc) shmeðwn. Se ut n thn perðptwsh lème epðshc ìti h P eðni leptìterh pì thn P.

56 Olokl rwm Riemnn (d) 'Estw P, P dôo dimerðseic tou [, b]. H koin eklèptunsh twn P, P eðni h dimèrish P = P P. EÔkol blèpoume ìti h P eðni dimèrish tou [, b] ki ìti n P eðni mi dimèrish leptìterh tìso pì thn P ìso ki pì thn P tìte P P (dhld, h P = P P eðni h mikrìterh dunt dimèrish tou [, b] pou ekleptônei tutìqron thn P ki thn P ). JewroÔme t r mi frgmènh sunˆrthsh f : [, b] R ki mi dimèrish P = { = x < x < < x n = b} tou [, b]. H P dimerðzei to [, b] st upodist mt [x, x ], [x, x ],..., [x k, x k+ ],..., [x n, x n ]. Gi kˆje k =,,..., n orðzoume touc prgmtikoôc rijmoôc (4..5) m k (f, P ) = m k = inf{f(x) : x k x x k+ } ki (4..6) M k (f, P ) = M k = sup{f(x) : x k x x k+ }. 'Oloi utoð oi rijmoð orðzonti klˆ: h f eðni frgmènh sto [, b], ˆr eðni frgmènh se kˆje upodiˆsthm [x k, x k+ ]. Gi kˆje k, to sônolo {f(x) : x k x x k+ } eðni mh kenì ki frgmèno uposônolo tou R, ˆr èqei supremum ki infimum. Gi kˆje dimèrish P tou [, b] orðzoume t r to ˆnw ki to kˆtw ˆjroism thc f wc proc thn P me ton ex c trìpo: n (4..7) U(f, P ) = M k (x k+ x k ) eðni to ˆnw ˆjroism thc f wc proc P, ki n (4..8) L(f, P ) = m k (x k+ x k ) eðni to kˆtw ˆjroism thc f wc proc P. Apì tic (4..7) ki (4..8) blèpoume ìti gi kˆje dimèrish P isqôei (4..9) L(f, P ) U(f, P ) foô m k M k ki x k+ x k >, k =,,..., n. Se sqèsh me to {embdìn} pou prospjoôme n orðsoume, prèpei n skeftìmste to kˆtw ˆjroism L(f, P ) sn mi prosèggish pì kˆtw ki to ˆnw ˆjroism U(f, P ) sn mi prosèggish pì pˆnw. J deðxoume ìti isqôei mi polô pio isqur nisìtht pì thn (4..9): Prìtsh 4... 'Estw f : [, b] R frgmènh sunˆrthsh ki èstw P, P dôo dimerðseic tou [, b]. Tìte, (4..) L(f, P ) U(f, P ). Prthr ste ìti h (4..9) eðni eidik perðptwsh thc (4..): rkeð n pˆroume P = P = P sthn Prìtsh 4... H pìdeixh thc Prìtshc 4.. j bsisteð sto ex c L mm.

4. O orismìs tou Drboux 57 L mm 4..3. 'Estw P = { = x < x < < x k < x k+ < < x n = b} ki x k < y < x k+ gi kˆpoio k =,,..., n. An P = P {y} = { = x < x < < x k < y < x k+ < < x n = b}, tìte (4..) L(f, P ) L(f, P ) U(f, P ) U(f, P ). Dhld, me thn prosj kh enìc shmeðou y sthn dimèrish P, to ˆnw ˆjroism thc f {mikrðnei} en to kˆtw ˆjroism thc f {megl nei}. Apìdeixh tou L mmtoc 4..3. Jètoume (4..) m () k = inf{f(x) : x k x y} ki (4..3) m () k = inf{f(x) : y x x k+ }. Tìte, m k m () k ki m k m () k (ˆskhsh: n A B tìte inf B inf A). Grˆfoume L(f, P ) = [m (x x ) + + m () k (y x k) + m () k (x k+ y) + +m n (x n x n )] [m (x x ) + + m k (y x k ) + m k (x k+ y) + +m n (x n x n )] = [m (x x ) + + m k (x k+ x k ) + + m n (x n x n )] = L(f, P ). 'Omoi deðqnoume ìti U(f, P ) U(f, P ). Apìdeixh thc Prìtshc 4... Gi n podeðxoume thn (4..) jewroôme thn koin eklèptunsh P = P P twn P ki P. H P prokôptei pì thn P me didoqik prosj kh pepersmènwn to pl joc shmeðwn. An efrmìsoume to L mm 4..3 pepersmènec to pl joc forèc, pðrnoume L(f, P ) L(f, P ). 'Omoi blèpoume ìti U(f, P ) U(f, P ). Apì thn ˆllh pleurˆ, L(f, P ) U(f, P ). Sunduˆzontc t prpˆnw, èqoume (4..4) L(f, P ) L(f, P ) U(f, P ) U(f, P ). JewroÔme t r t uposônol tou R { } (4..5) A(f) = L(f, P ) : P dimèrish tou [, b] ki (4..6) B(f) = { } U(f, Q) : Q dimèrish tou [, b]. Apì thn Prìtsh 4.. èqoume: gi kˆje A(f) ki kˆje b B(f) isqôei b (exhg ste gitð). 'Ar, sup A(f) inf B(f) (ˆskhsh). An loipìn orðsoume sn kˆtw olokl rwm thc f sto [, b] to { } (4..7) f(x)dx = sup L(f, P ) : P dimèrish tou [, b]

58 Olokl rwm Riemnn ki sn ˆnw olokl rwm thc f sto [, b] to (4..8) èqoume { } f(x)dx = inf U(f, Q) : Q dimèrish tou [, b], (4..9) f(x)dx f(x)dx. Orismìc 4..4. Mi frgmènh sunˆrthsh f : [, b] R lègeti Riemnn oloklhr simh n (4..) f(x)dx = I = f(x)dx. O rijmìc I (h koin tim tou kˆtw ki tou ˆnw oloklhr mtoc thc f sto [, b]) lègeti olokl rwm Riemnn thc f sto [, b] ki sumbolðzeti me (4..) f(x)dx f. 4. To krit rio oloklhrwsimìthtc tou Riemnn O orismìc tou oloklhr mtoc pou d sme sthn prohgoômenh prˆgrfo eðni dôsqrhstoc: den eðni eôkolo n ton qrhsimopoi sei kneðc gi n dei n mi frgmènh sunˆrthsh eðni oloklhr simh ìqi. Sun jwc, qrhsimopoioôme to kìloujo krit rio oloklhrwsimìthtc. Je rhm 4.. (krit rio tou Riemnn). 'Estw f : [, b] R frgmènh sunˆrthsh. H f eðni Riemnn oloklhr simh n ki mìno n gi kˆje ε > mporoôme n broôme dimèrish P ε tou [, b] ste (4..) U(f, P ε ) L(f, P ε ) < ε. Apìdeixh. Upojètoume pr t ìti h f eðni Riemnn oloklhr simh. Dhld, (4..) f(x)dx = f(x)dx = f(x)dx. 'Estw ε >. Apì ton orismì tou kˆtw oloklhr mtoc wc supremum tou A(f) ki pì ton ε-qrkthrismì tou supremum, upˆrqei dimèrish P = P (ε) tou [, b] ste (4..3) f(x)dx < L(f, P ) + ε. OmoÐwc, pì ton orismì tou ˆnw oloklhr mtoc, upˆrqei dimèrish P = P (ε) tou [, b] ste (4..4) f(x)dx > U(f, P ) ε.

4. To krit rio oloklhrwsimìthts tou Riemnn 59 JewroÔme thn koin eklèptunsh P ε = P P. Tìte, pì thn Prìtsh 4.. èqoume U(f, P ε ) ε U(f, P ) ε < f(x)dx = f(x)dx < L(f, P ) + ε L(f, P ε) + ε, p ìpou èpeti ìti (4..5) U(f, P ε ) L(f, P ε ) < ε. AntÐstrof: upojètoume ìti gi kˆje ε > upˆrqei dimèrish P ε tou [, b] ste (4..6) U(f, P ε ) < L(f, P ε ) + ε. Tìte, gi kˆje ε > èqoume f(x)dx U(f, P ε ) < L(f, P ε ) + ε Epeid to ε > tn tuqìn, èpeti ìti (4..7) f(x)dx f(x)dx, f(x)dx + ε. ki foô h ntðstrofh nisìtht isqôei pˆnt, h f eðni Riemnn oloklhr simh. To krit rio tou Riemnn ditup neti isodônm wc ex c (exhg ste gitð). Je rhm 4.. (krit rio tou Riemnn). 'Estw f : [, b] R frgmènh sunˆrthsh. H f eðni Riemnn oloklhr simh n ki mìno n upˆrqei koloujð {P n : n N} dimerðsewn tou [, b] ste (4..8) lim n ( U(f, Pn ) L(f, P n ) ) =. PrdeÐgmt. J qrhsimopoi soume to krit rio tou Riemnn gi n exetˆsoume n oi prkˆtw sunrt seic eðni Riemnn oloklhr simec: () H sunˆrthsh f : [, ] R me f(x) = x. Gi kˆje n N jewroôme th dimèrish P n tou [, ] se n Ðs upodist mt m kouc /n: { (4..9) P n = < n < n < < n < n }. n n = H sunˆrthsh f(x) = x eðni Ôxous sto [, ], epomènwc L(f, P n ) = f() ( ) ( ) n n + f n n + + f n n = ) ( + n n + (n ) + + n n = + + + (n ) (n )n(n ) n 3 = 6n 3 = n 3n + 6n = 3 n + 6n

6 Olokl rwm Riemnn ki 'Epeti ìti ( ) U(f, P n ) = f n n + f = ( n ( ) n n + n + + n n ( n n n) + + f n ) = + + + n n(n + )(n + ) n 3 = 6n 3 = n + 3n + 6n = 3 + n + 6n. (4..) U(f, P n ) L(f, P n ) = n. Apì to Je rhm 4.. sumperðnoume ìti h f eðni Riemnn oloklhr simh. M- poroôme mˆlist n broôme thn tim tou oloklhr mtoc. Gi kˆje n N, AfoÔ (4..) èpeti ìti (4..) Dhld, (4..3) 3 n + 6n = L(f, P n ) 3 n + 6n 3 x dx = U(f, P n ) = 3 + n + 6n. 3 ki x dx = x dx 3 + n + 6n 3, x dx 3. x dx = 3. (b) H sunˆrthsh u : [, ] R me u(x) = x. MporeÐte n qrhsimopoi sete thn koloujð dimerðsewn tou prohgoômenou prdeðgmtoc gi n deðxete ìti iknopoieðti to krit rio tou Riemnn. Sto Ðdio sumpèrsm ktl goume n qrhsimopoi soume mi diforetik koloujð dimerðsewn. Gi kˆje n N jewroôme th dimèrish (4..4) P n = { < H u eðni Ôxous sto [, ], epomènwc (4..5) L(u, P n ) = n (n ) < < < n n < n n = n ( ) k (k + ) n n k n }.

4. To krit rio oloklhrwsimìthts tou Riemnn 6 ki (4..6) U(u, P n ) = 'Epeti ìti n k + n ( (k + ) n ) k n. n ( k + U(u, P n ) L(u, P n ) = n k ) ( (k + ) n n = n ( ) (k + ) n n k n = n. ) k n Apì to Je rhm 4.. sumperðnoume ìti h u eðni Riemnn oloklhr simh. Af - noume sn ˆskhsh n deðxete ìti (4..7) lim n L(u, P n) = lim n U(u, P n) = 3. H sugkekrimènh epilog dimerðsewn pou kˆnme èqei to pleonèkthm ìti mporeðte eôkol n grˆyete t L(u, P n ) ki U(u, P n ) se kleist morf. Apì thn (4..7) èpeti ìti (4..8) x dx = 3. (g) H sunˆrthsh tou Dirichlet g : [, ] R me { n x rhtìc g(x) = n x ˆrrhtoc den eðni Riemnn oloklhr simh. 'Estw P = { = x < x < < x k < x k+ < < x n = } tuqoôs dimèrish tou [, ]. UpologÐzoume to kˆtw ki to ˆnw ˆjroism thc g wc proc thn P. Gi kˆje k =,,..., n upˆrqoun rhtìc q k ki ˆrrhtoc α k sto (x k, x k+ ). AfoÔ g(q k ) =, g(α k ) = ki g(x) sto [x k, x k+ ], sumperðnoume ìti m k = ki M k =. Sunep c, n n (4..9) L(g, P ) = m k (x k+ x k ) = (x k+ x k ) = ki n n (4..) U(g, P ) = M k (x k+ x k ) = (x k+ x k ) =. AfoÔ h P tn tuqoôs dimèrish tou [, ], pðrnoume (4..) g(x)dx = ki g(x)dx =.

6 Olokl rwm Riemnn 'Ar, h g den eðni Riemnn oloklhr simh. (d) H sunˆrthsh h : [, ] R me h(x) = { x n x rhtìc n x ˆrrhtoc den eðni Riemnn oloklhr simh. 'Estw P = { = x < x < < x k < x k+ < < x n = } tuqoôs dimèrish tou [, ]. Gi kˆje k =,,..., n upˆrqei ˆrrhtoc α k sto (x k, x k+ ). AfoÔ h(α k ) = ki h(x) sto [x k, x k+ ], sumperðnoume ìti m k =. Sunep c, (4..) L(h, P ) =. EpÐshc, upˆrqei rhtìc q k > (x k + x k+ )/ sto (x k, x k+ ), ˆr M k h(q k ) > (x k + x k+ )/. 'Epeti ìti U(h, P ) > n x k + x k+ (x k+ x k ) = n (x k+ x k) = x n x =. AfoÔ (4..3) U(h, P ) L(h, P ) > gi kˆje dimèrish P tou [, ], to krit rio tou Riemnn den iknopoieðti (pˆrte ε = /3). 'Ar, h h den eðni Riemnn oloklhr simh. (e) H sunˆrthsh w : [, ] R me { n x / Q x = w(x) = q n x = p q, p, q N, MKD(p, q) = eðni Riemnn oloklhr simh. EÔkol elègqoume ìti L(w, P ) = gi kˆje dimèrish P tou [, ]. 'Estw ε >. PrthroÔme ìti to sônolo A = {x [, ] : w(x) ε} eðni pepersmèno. [Prˆgmti, n w(x) ε tìte x = p/q ki w(x) = /q ε dhld q /ε. Oi rhtoð tou [, ] pou grˆfonti sn nˆgwg klˆsmt me pronomst to polô Ðso me [/ε] eðni pepersmènoi to pl joc (èn ˆnw frˆgm gi to pl joc touc eðni o rijmìc + + + [/ε] exhg ste gitð)]. 'Estw z < z < < z N mð rðjmhsh twn stoiqeðwn tou A. MporoÔme n broôme xèn upodist mt [ i, b i ] tou [, ] pou èqoun m kh b i i < ε/n ki iknopoioôn t ex c: >, i < z i < b i n i < N ki N < z N b N (prthr ste ìti n ε tìte z N = opìte prèpei n epilèxoume b N = ). An jewr soume th dimèrish (4..4) P ε = { < < b < < b < < N < b N },

4.3 DÔo klˆseis Riemnn oloklhr simwn sunrt sewn 63 èqoume U(w, P ε ) ε ( ) + (b ) + ε ( b ) + + (b N N ) +ε ( N b N ) + (b N N ) + ε ( b N ) ( ) ε + ( b ) + + ( N b N ) + ( b N ) + < ε. N (b i i ) i= Gi to tuqìn ε > br kme dimèrish P ε tou [, ] me thn idiìtht (4..5) U(w, P ε ) L(w, P ε ) < ε. Apì to Je rhm 4.., h w eðni Riemnn oloklhr simh. 4.3 DÔo klˆseic Riemnn oloklhr simwn sunrt sewn Qrhsimopoi ntc to krit rio tou Riemnn (Je rhm 4..) j deðxoume ìti oi monìtonec ki oi suneqeðc sunrt seic f : [, b] R eðni Riemnn oloklhr simec. Je rhm 4.3.. Kˆje monìtonh sunˆrthsh f : [, b] R eðni Riemnn oloklhr simh. Apìdeixh. QwrÐc periorismì thc genikìthtc upojètoume ìti h f eðni Ôxous. H f eðni profn c frgmènh: gi kˆje x [, b] èqoume (4.3.) f() f(x) f(b). 'Ar, èqei nìhm n exetˆsoume thn Ôprxh oloklhr mtoc gi thn f. 'Estw ε >. J broôme n N rketˆ megˆlo ste gi th dimèrish { (4.3.) P n =, + b } (b ) n(b ), +,..., + = b n n n tou [, b] se n Ðs upodist mt n isqôei (4.3.3) U(f, P n ) L(f, P n ) < ε. Jètoume (4.3.4) x k = + Tìte, foô h f eðni Ôxous èqoume U(f, P n ) = k(b ), k =,,..., n. n n n M k (x k+ x k ) = f(x k+ ) b n = b n (f(x ) + + f(x n )),

64 Olokl rwm Riemnn en 'Ar, L(f, P n ) = n n m k (x k+ x k ) = f(x k ) b n = b n (f(x ) + + f(x n )). (4.3.5) U(f, P n ) L(f, P n ) = [f(x n) f(x )](b ) n = [f(b) f()](b ), n to opoðo gðneti mikrìtero pì to ε > pou mc dìjhke, rkeð to n n eðni rketˆ megˆlo. Apì to Je rhm 4.., h f eðni Riemnn oloklhr simh. Je rhm 4.3.. Kˆje suneq c sunˆrthsh f : [, b] R eðni Riemnn oloklhr simh. Apìdeixh. 'Estw ε >. H f eðni suneq c sto kleistì diˆsthm [, b], ˆr eðni omoiìmorf suneq c. MporoÔme loipìn n broôme δ > me thn ex c idiìtht: An x, y [, b] ki x y < δ, tìte f(x) f(y) < MporoÔme epðshc n broôme n N ste ε. b (4.3.6) b n < δ. QwrÐzoume to [, b] se n upodist mt tou idðou m kouc b n. JewroÔme dhld th dimèrish { (4.3.7) P n = OrÐzoume, + b n (4.3.8) x k = + } (b ) n(b ), +,..., + = b. n n k(b ), k =,,..., n. n 'Estw k =,,..., n. H f eðni suneq c sto kleistì diˆsthm [x k, x k+ ], ˆr pðrnei mègisth ki elˆqisth tim se utì. Upˆrqoun dhld y k, y k [x k, x k+ ] ste (4.3.9) M k = f(y k) ki m k = f(y k). Epiplèon, to m koc tou [x k, x k+ ] eðni Ðso me b n (4.3.) y k y k < δ. Apì thn epilog tou δ pðrnoume < δ, ˆr (4.3.) M k m k = f(y k) f(y k) = f(y k) f(y k) < ε b.

4.4 Idiìthtes tou oloklhr mtos Riemnn 65 'Epeti ìti U(f, P n ) L(f, P n ) = < n (M k m k )(x k+ x k ) n ε b (x k+ x k ) ε = (b ) = ε. b Apì to Je rhm 4.., h f eðni Riemnn oloklhr simh. 4.4 Idiìthtec tou oloklhr mtoc Riemnn Se ut n thn prˆgrfo podeiknôoume usthrˆ merikèc pì tic pio bsikèc idiìthtec tou oloklhr mtoc Riemnn. Oi podeðxeic twn upoloðpwn eðni mi kl ˆskhsh pou j sc bohj sei n exoikeiwjeðte me tic dimerðseic, t ˆnw ki kˆtw jroðsmt klp. Je rhm 4.4.. An f(x) = c gi kˆje x [, b], tìte (4.4.) f(x)dx = c(b ). Apìdeixh: 'Estw P = { = x < x < < x n = b} mi dimèrish tou [, b]. Gi kˆje k =,,..., n èqoume m k = M k = c. 'Ar, n (4.4.) L(f, P ) = U(f, P ) = c(x k+ x k ) = c(b ). 'Epeti ìti (4.4.3) 'Ar, (4.4.4) f(x)dx = c(b ) = f(x)dx. f(x)dx = c(b ). Je rhm 4.4.. 'Estw f, g : [, b] R oloklhr simec sunrt seic. Tìte, h f + g eðni oloklhr simh ki (4.4.5) [f(x) + g(x)]dx = f(x)dx + g(x)dx. Apìdeixh. 'Estw P = { = x < x < < x n = b} dimèrish tou [, b]. Gi kˆje k =,,..., n orðzoume m k = inf{(f + g)(x) : x k x x k+ } M k = sup{(f + g)(x) : x k x x k+ } m k = inf{f(x) : x k x x k+ } M k = sup{f(x) : x k x x k+ } m k = inf{g(x) : x k x x k+ } M k = sup{g(x) : x k x x k+ }.

66 Olokl rwm Riemnn Gi kˆje x [x k, x k+ ] èqoume m k + m k f(x) + g(x). 'Ar, (4.4.6) m k + m k m k. OmoÐwc, gi kˆje x [x k, x k+ ] èqoume M k + M k (4.4.7) M k + M k M k. 'Epeti ìti f(x) + g(x). 'Ar, (4.4.8) L(f, P ) + L(g, P ) L(f + g, P ) U(f + g, P ) U(f, P ) + U(g, P ). 'Estw ε >. Upˆrqoun dimerðseic P, P tou [, b] ste (4.4.9) U(f, P ) ε < f(x)dx < L(f, P ) + ε ki (4.4.) U(g, P ) ε < g(x)dx < L(g, P ) + ε. An jewr soume thn koin touc eklèptunsh P = P P èqoume U(f, P ) + U(g, P ) ε U(f, P ) + U(g, P ) ε Sunduˆzontc me thn (4.4.8) blèpoume ìti < (f + g)(x)dx ε U(f + g, P ) ε AfoÔ to ε > tn tuqìn, f(x)dx + g(x)dx < L(f, P ) + L(g, P ) + ε L(f, P ) + L(g, P ) + ε. L(f + g, P ) + ε f(x)dx + (f + g)(x)dx + ε. g(x)dx (4.4.) 'Omwc, (4.4.) 'Ar, (f + g)(x)dx f(x)dx + g(x)dx (f + g)(x)dx (f + g)(x)dx. (f + g)(x)dx. (4.4.3) 'Epeti to Je rhm. (f + g)(x)dx = f(x)dx + g(x)dx = (f + g)(x)dx.

4.4 Idiìthtes tou oloklhr mtos Riemnn 67 Je rhm 4.4.3. 'Estw f : [, b] R oloklhr simh ki èstw t R. Tìte, h tf eðni oloklhr simh sto [, b] ki (4.4.4) (tf)(x)dx = t f(x)dx. Apìdeixh. Ac upojèsoume pr t ìti t >. 'Estw P = { = x < x < < x n = b} dimèrish tou [, b]. An gi k =,,..., n orðsoume (4.4.5) m k = inf{(tf)(x) : x k x x k+ }, M k = sup{(tf)(x) : x k x x k+ } ki (4.4.6) m k = inf{f(x) : x k x x k+ }, M k = sup{f(x) : x k x x k+ }, eðni fnerì ìti (4.4.7) m k = tm k ki M k = tm k. 'Ar, (4.4.8) L(tf, P ) = tl(f, P ) ki U(tf, P ) = tu(f, P ). 'Epeti ìti (4.4.9) (tf)(x)dx = t AfoÔ h f eðni oloklhr simh, èqoume f(x)dx ki (tf)(x)dx = t f(x)dx. (4.4.) f(x)dx = 'Epeti ìti h tf eðni Riemnn oloklhr simh, ki (4.4.) (tf)(x)dx = t f(x)dx. f(x)dx. An t <, h mình llg sto prohgoômeno epiqeðrhm eðni ìti t r m k = tm k ki M k = tm k. Sumplhr ste thn pìdeixh mìnoi sc. Tèloc, n t = èqoume tf. 'Ar, (4.4.) tf = = f. Apì t Jewr mt 4.4. ki 4.4.3 prokôptei ˆmes h {grmmikìtht tou oloklhr mtoc}. Je rhm 4.4.4 (grmmikìtht tou oloklhr mtoc). An f, g : [, b] R eðni dôo oloklhr simec sunrt seic ki t, s R, tìte h tf + sg eðni oloklhr simh sto [, b] ki (4.4.3) (tf + sg)(x)dx = t f(x)dx + s g(x)dx.

68 Olokl rwm Riemnn Je rhm 4.4.5. 'Estw f : [, b] R frgmènh sunˆrthsh ki èstw c (, b). H f eðni oloklhr simh sto [, b] n ki mìno n eðni oloklhr simh st [, c] ki [c, b]. Tìte, isqôei (4.4.4) f(x)dx = c f(x)dx + c f(x)dx. Apìdeixh. Upojètoume pr t ìti h f eðni oloklhr simh st [, c] ki [c, b]. 'Estw ε >. Upˆrqoun dimerðseic P tou [, c] ki P tou [c, b] ste (4.4.5) L(f, P ) ki (4.4.6) L(f, P ) c c f(x)dx U(f, P ) ki U(f, P ) L(f, P ) < ε f(x)dx U(f, P ) ki U(f, P ) L(f, P ) < ε. To sônolo P ε = P P eðni dimèrish tou [, b] ki isqôoun oi (4.4.7) L(f, P ε ) = L(f, P ) + L(f, P ) ki U(f, P ε ) = U(f, P ) + U(f, P ). Apì tic prpˆnw sqèseic pðrnoume U(f, P ε ) L(f, P ε ) = (U(f, P ) L(f, P )) + (U(f, P ) L(f, P )) < ε + ε = ε. AfoÔ to ε > tn tuqìn, h f eðni oloklhr simh sto [, b] (krit rio tou Riemnn). Epiplèon, gi thn P ε èqoume (4.4.8) L(f, P ε ) ki, pì tic (4.4.5), (4.4.6) ki (4.4.7), (4.4.9) L(f, P ε ) Epomènwc, (4.4.3) c f(x)dx + ( c f(x)dx f(x)dx + ki foô to ε > tn tuqìn, (4.4.3) f(x)dx = c f(x)dx U(f, P ε ) c c f(x)dx U(f, P ε ). f(x)dx) f(x)dx + c U(f, P ε) L(f, P ε ) < ε, f(x)dx. AntÐstrof: upojètoume ìti h f eðni oloklhr simh sto [, b] ki jewroôme ε >. Upˆrqei dimèrish P tou [, b] ste (4.4.3) U(f, P ) L(f, P ) < ε.

4.4 Idiìthtes tou oloklhr mtos Riemnn 69 An c / P jètoume P = P {c}, opìte pˆli èqoume (4.4.33) U(f, P ) L(f, P ) U(f, P ) L(f, P ) < ε. MporoÔme loipìn n upojèsoume ìti c P. OrÐzoume P = P [, c] ki P = P [c, b]. Oi P, P eðni dimerðseic twn [, c] ki [c, b] ntðstoiq, ki (4.4.34) L(f, P ) = L(f, P ) + L(f, P ), U(f, P ) = U(f, P ) + U(f, P ). AfoÔ (4.4.35) (U(f, P ) L(f, P )) + (U(f, P ) L(f, P )) = U(f, P ) L(f, P ) < ε, èpeti ìti (4.4.36) U(f, P ) L(f, P ) < ε ki U(f, P ) L(f, P ) < ε. AfoÔ to ε > tn tuqìn, to krit rio tou Riemnn deðqnei ìti h f eðni oloklhr simh st [, c] ki [c, b]. T r, pì to pr to mèroc thc pìdeixhc pðrnoume thn isìtht (4.4.37) f(x)dx = c f(x)dx + c f(x)dx. Je rhm 4.4.6. 'Estw f : [, b] R oloklhr simh sunˆrthsh. Upojètoume ìti m f(x) M gi kˆje x [, b]. Tìte, (4.4.38) m(b ) ShmeÐwsh. O rijmìc (4.4.39) eðni h mèsh tim thc f sto [, b]. b f(x)dx M(b ). f(x)dx Apìdeixh. ArkeÐ n dipist sete ìti gi kˆje dimèrish P tou [, b] isqôei (4.4.4) m(b ) L(f, P ) U(f, P ) M(b ) (to opoðo eðni polô eôkolo). Pìrism 4.4.7. () 'Estw f : [, b] R oloklhr simh sunˆrthsh. Upojètoume ìti f(x) gi kˆje x [, b]. Tìte, (4.4.4) f(x)dx. (b) 'Estw f, g : [, b] R oloklhr simec sunrt seic. Upojètoume ìti f(x) g(x) gi kˆje x [, b]. Tìte, (4.4.4) f(x)dx g(x)dx.

7 Olokl rwm Riemnn Apìdeixh. () Efrmìzoume to Je rhm 4.4.6: mporoôme n pˆroume m =. (b) H f g eðni oloklhr simh sunˆrthsh ki (f g)(x) gi kˆje x [, b]. Efrmìzoume to () gi thn f g ki qrhsimopoioôme th grmmikìtht tou oloklhr mtoc. Je rhm 4.4.8. 'Estw f : [, b] [m, M] oloklhr simh sunˆrthsh ki èstw φ : [m, M] R suneq c sunˆrthsh. Tìte, h φ f : [, b] R eðni oloklhr simh. Apìdeixh. 'Estw ε >. J broôme dimèrish P tou [, b] me thn idiìtht U(φ f, P ) L(φ f, P ) < ε. To zhtoômeno èpeti pì to krit rio tou Riemnn. H φ eðni suneq c sto [m, M], ˆr eðni frgmènh: upˆrqei A > ste φ(ξ) A gi kˆje ξ [m, M]. EpÐshc, h φ eðni omoiìmorf suneq c: n jèsoume ε = ε/(a + b ) >, upˆrqei < δ < ε ste, gi kˆje ξ, η [m, M] me ξ η < δ isqôei φ(ξ) φ(η) < ε. Efrmìzontc to krit rio tou Riemnn gi thn oloklhr simh sunˆrthsh f, brðskoume dimerish P = { = x < x < < x k < x k+ < < x n = b} ste n (4.4.43) U(f, P ) L(f, P ) = (M k (f) m k (f))(x k+ x k ) < δ. OrÐzoume PrthroÔme t ex c: I = { k n : M k (f) m k (f) < δ} J = { k n : M k (f) m k (f) δ}. (i) An k I, tìte gi kˆje x, x [x k, x k+ ] èqoume f(x) f(x ) M k (f) m k (f) < δ. PÐrnontc ξ = f(x) ki η = f(x ), èqoume ξ, η [m, M] ki ξ η < δ. 'Ar, (φ f)(x) (φ f)(x ) = φ(ξ) φ(η) < ε. AfoÔ t x, x tn tuqìnt sto [x k, x k+ ], sumperðnoume ìti M k (φ f) m k (φ f) ε (exhg ste gitð). 'Epeti ìti (4.4.44) k I (M k (φ f) m k (φ f))(x k+ x k ) ε k I(x k+ x k ) (b )ε. (ii) Gi to J èqoume, pì thn (4.4.43), (4.4.45) δ k J (x k+ x k ) k J(M k (f) m k (f))(x k+ x k ) < δ, ˆr (4.4.46) (x k+ x k ) < δ < ε. k J EpÐshc, (4.4.47) (φ f)(x) (φ f)(x ) (φ f)(x) + (φ f)(x ) A

4.4 Idiìthtes tou oloklhr mtos Riemnn 7 gi kˆje x, x [x k, x k+ ], ˆr M k (φ f) m k (φ f) A gi kˆje k J. 'Epeti ìti (4.4.48) k (φ f) m k (φ f))(x k+ x k ) A k J(M (x k+ x k ) < Aε. k J Apì tic (4.4.44) ki (4.4.48) sumperðnoume ìti U(φ f, P ) L(φ f, P ) = Autì oloklhr nei thn pìdeixh. n (M k (φ f) m k (φ f))(x k+ x k ) = k I(M k (φ f) m k (φ f))(x k+ x k ) + k J(M k (φ f) m k (φ f))(x k+ x k ) < (b )ε + Aε = ε. Qrhsimopoi ntc to Je rhm 4.4.8 mporoôme n elègxoume eôkol thn oloklhrwsimìtht difìrwn sunrt sewn pou prokôptoun pì thn sônjesh mic oloklhr simhc sunˆrthshc f me ktˆllhlec suneqeðc sunrt seic. Je rhm 4.4.9. 'Estw f, g : [, b] R oloklhr simec sunrt seic. Tìte, () h f eðni oloklhr simh ki (4.4.49) (b) h f eðni oloklhr simh. (g) h fg eðni oloklhr simh. f(x)dx f(x) dx. Apìdeixh. T () ki (b) eðni ˆmesec sunèpeiec tou Jewr mtoc 4.4.8. Gi to (g) grˆyte (4.4.5) fg = (f + g) (f g) ki qrhsimopoi ste to (b) se sundusmì me to gegonìc ìti oi f + g, f g eðni oloklhr simec. Mi sômbsh. Wc t r orðsme to f(x)dx mìno sthn perðptwsh < b (douleôme sto kleistì diˆsthm [, b]). Gi prktikoôc lìgouc epekteðnoume ton orismì ki sthn perðptwsh b wc ex c: () n = b, jètoume f = (gi kˆje f). (b) n > b ki h f : [b, ] R eðni oloklhr simh, orðzoume 4 (4.4.5) f(x)dx = b f(x)dx.

7 Olokl rwm Riemnn 4.5 O orismìc tou Riemnn* O orismìc pou d sme gi thn oloklhrwsimìtht mic frgmènhc sunˆrthshc f : [, b] R ofeðleti ston Drboux. O pr toc usthrìc orismìc thc oloklhrwsimìthtc dìjhke pì ton Riemnn ki eðni o ex c: Orismìc 4.5.. 'Estw f : [, b] R frgmènh sunˆrthsh. Lème ìti h f eðni oloklhr simh sto [, b] n upˆrqei ènc prgmtikìc rijmìc I(f) me thn ex c idiìtht: Gi kˆje ε > mporoôme n broôme δ > ste: n P = { = x < x < < x n = b} eðni dimèrish tou [, b] me plˆtoc P < δ ki n ξ k [x k, x k+ ], k =,,..., n eðni tuqoôs epilog shmeðwn pì t upodist mt pou orðzei h P, tìte n f(ξ k )(x k+ x k ) I(f) < ε. Se ut thn perðptwsh lème ìti o I(f) eðni to (R)-olokl rwm thc f sto [, b]. Sumbolismìc. Sun jwc grˆfoume Ξ gi thn epilog shmeðwn {ξ, ξ,..., ξ n } ki (f, P, Ξ) gi to ˆjroism (4.5.) n f(ξ k )(x k+ x k ). Prthr ste ìti t r to Ξ {upeisèrqeti} sto sumbolismì (f, P, Ξ) foô gi thn Ðdi dimèrish P mporoôme n èqoume pollèc diforetikèc epilogèc Ξ = {ξ, ξ,..., ξ n } me ξ k [x k, x k+ ]. (4.5.) H bsik idè pðsw pì ton orismì eðni ìti f(x)dx = lim (f, P, Ξ) ìtn to plˆtoc thc P teðnei sto mhdèn ki t ξ k epilègonti ujðret st upodist mt pou orðzei h P. Epeid den èqoume sunnt sei tètoiou eðdouc {ìri} wc t r, ktfeôgoume ston {eyilontikì orismì}. Skopìc ut c thc prgrˆfou eðni h pìdeixh thc isodunmðc twn dôo orism n oloklhrwsimìthtc: Je rhm 4.5.. 'Estw f : [, b] R frgmènh sunˆrthsh. H f eðni oloklhr simh ktˆ Drboux n ki mìno n eðni oloklhr simh ktˆ Riemnn. Apìdeixh. Upojètoume pr t ìti h f eðni oloklhr simh ktˆ Riemnn. Grˆfoume I(f) gi to olokl rwm thc f me ton orismì tou Riemnn. 'Estw ε >. MporoÔme n broôme mi dimèrish P = { = x < x < < x n = b} (me rketˆ mikrì plˆtoc) ste gi kˆje epilog shmeðwn Ξ = {ξ, ξ,..., ξ n } me ξ k [x k, x k+ ] n isqôei (4.5.3) n f(ξ k )(x k+ x k ) I(f) < ε 4.

4.5 O orismìs tou Riemnn* 73 Gi kˆje k =,,..., n mporoôme n broôme ξ k, ξ k [x k, x k+ ] ste (4.5.4) m k > f(ξ k) 'Ar, ε 4(b ) ki M k < f(ξ k ε ) + 4(b ). n (4.5.5) L(f, P ) > f(ξ k)(x k+ x k ) ε 4 > I(f) ε ki n (4.5.6) U(f, P ) < f(ξ k )(x k+ x k ) + ε 4 < I(f) + ε. 'Epeti ìti (4.5.7) U(f, P ) L(f, P ) < ε, dhld h f eðni oloklhr simh ktˆ Drboux. EpÐshc, (4.5.8) I(f) ε < f(x)dx ki foô to ε > tn tuqìn, f(x)dx < I(f) + ε, (4.5.9) Dhld, f(x)dx = f(x)dx = I(f). (4.5.) f(x)dx = I(f). AntÐstrof: upojètoume ìti h f eðni oloklhr simh me ton orismì tou Drboux. 'Estw ε >. Upˆrqei dimèrish P = { = x < x < < x n = b} tou [, b] ste (4.5.) U(f, P ) L(f, P ) < ε 4. H f eðni frgmènh, dhld upˆrqei M > ste f(x) M gi kˆje x [, b]. Epilègoume (4.5.) δ = ε 6nM >. 'Estw P dimèrish tou [, b] me plˆtoc P < δ, h opoð eðni ki eklèptunsh thc P. Tìte, gi kˆje epilog Ξ shmeðwn pì t upodist mt pou orðzei h P èqoume f(x)dx ε 4 < L(f, P ) L(f, P ) (f, P, Ξ) U(f, P ) U(f, P ) < f(x)dx + ε 4.

74 Olokl rwm Riemnn Dhld, (4.5.3) (f, P, Ξ) f(x)dx < ε. Zhtˆme n deðxoume to Ðdio prˆgm gi tuqoôs dimèrish P me plˆtoc mikrìtero pì δ (h duskolð eðni ìti mi tètoi dimèrish den èqei knèn lìgo n eðni eklèptunsh thc P ). 'Estw P = { = y < y < < y m = b} mi tètoi dimèrish tou [, b]. J {prosjèsoume} sthn P èn-èn ìl t shmeð x k thc P t opoð den n koun sthn P (utˆ eðni to polô n ). Ac poôme ìti èn tètoio x k brðsketi nˆmes st didoqikˆ shmeð y l < y l+ thc P. JewroÔme thn P = P {x k } ki tuqoôs epilog Ξ () = {ξ, ξ,..., ξ m } me ξ l [y l, y l+ ], l =,,..., m. Epilègoume dôo shmeð ξ l [y l, x k ] ki ξ l [x k, y l+ ] ki jewroôme thn epilog shmeðwn Ξ () = {ξ, ξ,..., ξ l, ξ l, ξ l,..., ξ m } pou ntistoiqeð sthn P. 'Eqoume (f, P, Ξ () ) (f, P, Ξ () ) = f(ξ l )(y l+ y l ) f(ξ l)(x k y l ) f(ξ l )(y l+ x k ) 3M mx y l+ y l < 3Mδ l = ε n. Antikjist ntc th dosmènh (P, Ξ () ) me ìlo ki leptìterec dimerðseic (P k, Ξ (k) ) pou prokôptoun me thn prosj kh shmeðwn thc P, metˆ pì n to polô b mt ftˆnoume se mi dimèrish P ki mi epilog shmeðwn Ξ () me tic ex c idiìthtec: () h P eðni koin eklèptunsh twn P ki P, ki èqei plˆtoc mikrìtero pì δ. (b) foô h P eðni eklèptunsh thc P, ìpwc sthn (4.5.3) èqoume (4.5.4) (f, P, Ξ () ) f(x)dx < ε. (g) foô kˆnme to polô n b mt gi n ftˆsoume sthn P ki foô se kˆje b m t jroðsmt peðqn to polô ε, èqoume n (f, P, Ξ () ) (f, P, Ξ () ) < n ε n = ε. Dhld, gi thn tuqoôs dimèrish P plˆtouc < δ ki gi thn tuqoôs epilog Ξ () shmeðwn pì t upodist mt thc P, èqoume (f, P, Ξ () ) f(x)dx < (f, P, Ξ () ) (f, P, Ξ () ) + (f, P, Ξ () ) f(x)dx < ε + ε = ε. 'Epeti ìti h f eðni oloklhr simh me ton orismì tou Riemnn, kj c ki ìti oi I(f) ki f(x)dx eðni Ðsoi.

4.6 Ask seic Omˆd A'. Erwt seic ktnìhshc 4.6 Ask seis 75 'Estw f : [, b] R. Exetˆste n oi prkˆtw protˆseic eðni lhjeðc yeudeðc (itiolog ste pl rwc thn pˆnths sc).. An h f eðni Riemnn oloklhr simh, tìte h f eðni frgmènh.. An h f eðni Riemnn oloklhr simh, tìte pðrnei mègisth tim. 3. An h f eðni frgmènh, tìte eðni Riemnn oloklhr simh. 4. An h f eðni Riemnn oloklhr simh, tìte h f eðni Riemnn oloklhr simh. 5. An h f eðni Riemnn oloklhr simh, tìte upˆrqei c [, b] ste f(c)(b ) = f(x) dx. 6. An h f eðni frgmènh ki n L(f, P ) = U(f, P ) gi kˆje dimèrish P tou [, b], tìte h f eðni stjer. 7. An h f eðni frgmènh ki n upˆrqei dimèrish P ste L(f, P ) = U(f, P ), tìte h f eðni Riemnn oloklhr simh. 8. An h f eðni Riemnn oloklhr simh ki n f(x) = gi kˆje x [, b] Q, tìte f(x)dx =. Omˆd B' 9. 'Estw f : [, ] R frgmènh sunˆrthsh me thn idiìtht: gi kˆje < b h f eðni oloklhr simh sto diˆsthm [b, ]. DeÐxte ìti h f eðni oloklhr simh sto [, ].. ApodeÐxte ìti h sunˆrthsh f : [, ] R me f(x) = sin x f() = eðni oloklhr simh. n x ki. 'Estw g : [, b] R frgmènh sunˆrthsh. Upojètoume ìti h g eðni suneq c pntoô, ektìc pì èn shmeðo x (, b). DeÐxte ìti h g eðni oloklhr simh.. Qrhsimopoi ntc to krit rio tou Riemnn podeðxte ìti oi prkˆtw sunrt - seic eðni oloklhr simec: () f : [, ] R me f(x) = x. (b) f : [, π/] R me f(x) = sin x. 3. Exetˆste n oi prkˆtw sunrt seic eðni oloklhr simec sto [, ] ki upologðste to olokl rwm touc (n upˆrqei): () f(x) = x + [x].

76 Olokl rwm Riemnn (b) f(x) = n x = k gi kˆpoion k N, ki f(x) = lli c. 4. 'Estw f : [, b] R suneq c sunˆrthsh me f(x) gi kˆje x [, b]. DeÐxte ìti f(x)dx = n ki mìno n f(x) = gi kˆje x [, b]. 5. 'Estw f, g : [, b] R suneqeðc sunrt seic ste f(x)dx = DeÐxte ìti upˆrqei x [, b] ste f(x ) = g(x ). g(x)dx. 6. 'Estw f : [, b] R suneq c sunˆrthsh me thn idiìtht: gi kˆje suneq sunˆrthsh g : [, b] R isqôei DeÐxte ìti f(x) = gi kˆje x [, b]. f(x)g(x)dx =. 7. 'Estw f : [, b] R suneq c sunˆrthsh me thn idiìtht: gi kˆje suneq sunˆrthsh g : [, b] R pou iknopoieð thn g() = g(b) =, isqôei DeÐxte ìti f(x) = gi kˆje x [, b]. f(x)g(x)dx =. 8. 'Estw f, g : [, b] R oloklhr simec sunrt seic. DeÐxte thn nisìtht Cuchy-Schwrz: ( f(x)g(x)dx) ( ) f (x)dx ( ) g (x)dx. 9. 'Estw f : [, ] R oloklhr simh sunˆrthsh. DeÐxte ìti ( f(x)dx) f (x)dx. IsqÔei to Ðdio n ntiktst soume to [, ] me tuqìn diˆsthm [, b]?. 'Estw f : [, + ) R suneq c sunˆrthsh. DeÐxte ìti lim x + x x f(t)dt = f().

4.6 Ask seis 77. 'Estw f : [, ] R oloklhr simh sunˆrthsh. DeÐxte ìti h koloujð n = n n f ( ) k n sugklðnei sto f(x)dx. [Upìdeixh: Qrhsimopoi ste ton orismì tou Riemnn.]. DeÐxte ìti lim n + + + n n = n 3. 3. 'Estw f : [, ] R suneq c sunˆrthsh. OrÐzoume mi koloujð ( n ) jètontc n = f(xn )dx. DeÐxte ìti n f(). 4. DeÐxte ìti h koloujð γ n = + + 3 + + n n 5. 'Estw f : [, ] R Lipschitz suneq c sunˆrthsh ste gi kˆje x, y [, ]. DeÐxte ìti gi kˆje n N. f(x) f(y) M x y f(x)dx n n f ( ) k M n n xdx sugklðnei. Omˆd G' 6. 'Estw f : [, b] R gnhsðwc Ôxous ki suneq c sunˆrthsh. DeÐxte ìti f(x)dx = bf(b) f() f(b) f() f (x)dx. 7. 'Estw f : [, + ) [, + ) gnhsðwc Ôxous, suneq c ki epð sunˆrthsh me f() =. DeÐxte ìti gi kˆje, b > b me isìtht n ki mìno n f() = b. f(x)dx + f (x)dx 8. 'Estw f : [, b] R suneq c sunˆrthsh me thn ex c idiìtht: upˆrqei M > ste f(x) M x f(t) dt gi kˆje x [, b]. DeÐxte ìti f(x) = gi kˆje x [, b].

78 Olokl rwm Riemnn 9. 'Estw R. DeÐxte ìti den upˆrqei jetik suneq c sunˆrthsh f : [, ] R ste f(x)dx =, xf(x)dx = ki x f(x)dx =. 3. 'Estw f : [, b] R suneq c, mh rnhtik sunˆrthsh. Jètoume M = mx{f(x) : x [, b]}. DeÐxte oti h koloujð sugklðnei, ki lim n γ n = M. ( γ n = [f(x)] n dx ) /n 3. 'Estw f : [, b] R oloklhr simh sunˆrthsh. Skopìc ut c thc ˆskhshc eðni n deðxoume ìti h f èqei pollˆ shmeð sunèqeic. () Upˆrqei dimèrish P tou [, b] ste U(f, P ) L(f, P ) < b (exhg ste gitð). DeÐxte ìti upˆrqoun < b sto [, b] ste b < ki sup{f(x) : x b } inf{f(x) : x b } <. (b) Epgwgikˆ orðste kibwtismèn dist mt [ n, b n ] ( n, b n ) me m koc mikrìtero pì /n ste sup{f(x) : n x b n } inf{f(x) : n x b n } < n. (g) H tom ut n twn kibwtismènwn disthmˆtwn perièqei krib c èn shmeðo. DeÐxte ìti h f eðni suneq c se utì. (d) T r deðxte ìti h f èqei ˆpeir shmeð sunèqeic sto [, b] (den qreiˆzeti perissìterh douleiˆ!). 3. 'Estw f : [, b] R oloklhr simh (ìqi ngkstikˆ suneq c) sunˆrthsh me f(x) > gi kˆje x [, b]. DeÐxte ìti f(x)dx >. Omˆd D'. Sumplhr mt thc JewrÐc ApodeÐxte tic prkˆtw protˆseic. 33. 'Estw f, g, h : [, b] R treðc sunrt seic pou iknopoioôn thn f(x) g(x) h(x) gi kˆje x [, b]. Upojètoume ìti oi f, h eðni oloklhr simec ki f(x)dx = DeÐxte ìti h g eðni oloklhr simh ki g(x)dx = I. h(x)dx = I.

4.6 Ask seis 79 34. 'Estw f : [, b] R oloklhr simh sunˆrthsh. DeÐxte ìti h f eðni oloklhr simh. OmoÐwc, ìti h f eðni oloklhr simh. 35. 'Estw f, g : [, b] R oloklhr simec sunrt seic. DeÐxte ìti h f g eðni oloklhr simh. 36. 'Estw f : [, b] R oloklhr simh. DeÐxte ìti f(x)dx f(x) dx. 37. 'Estw f : R R sunˆrthsh oloklhr simh se kˆje kleistì diˆsthm thc morf c [, b]. DeÐxte ìti: () f(x)dx = f( x)dx. (b) f(x)dx = f( + b x)dx. (g) f(x)dx = +c f(x c)dx. +c (d) cb c f(t)dt = c f(ct)dt. (e) f(x)dx = n h f eðni peritt. (st) f(x)dx = f(x)dx n h f eðni ˆrti. 38. 'Estw f : [, b] R frgmènh sunˆrthsh. () DeÐxte ìti h f eðni oloklhr simh n ki mìno n gi kˆje ε > mporoôme n broôme klimkwtèc sunˆrt seic g ε, h ε : [, b] R me g ε f h ε ki h ε (x)dx g ε (x)dx < ε. (b) DeÐxte ìti h f eðni oloklhr simh n ki mìno n gi kˆje ε > mporoôme n broôme suneqeðc sunrt seic g ε, h ε : [, b] R me g ε f h ε ki h ε (x)dx g ε (x)dx < ε.

Kefˆlio 5 To jemeli dec je rhm tou ApeirostikoÔ LogismoÔ Se utì to Kefˆlio j lème ìti mi sunˆrthsh f : [, b] R eðni prgwgðsimh sto [, b] n h prˆgwgoc f (x) upˆrqei gi kˆje x (, b) ki, epiplèon, upˆrqoun oi pleurikèc prˆgwgoi f +() = lim x + f(x) f() x ki f (b) f(x) f(b) = lim. x b x b SumfwnoÔme n grˆfoume f () = f +() ki f (b) = f (b). 5. To je rhm mèshc tou OloklhrwtikoÔ LogismoÔ 'Estw f : [, b] R mi Riemnn oloklhr simh sunˆrthsh. Sto prohgoômeno Kefˆlio orðsme th mèsh tim (5..) b f(x)dx thc f sto [, b]. An h f upotejeð suneq c, tìte upˆrqei ξ [, b] me thn idiìtht (5..) f(ξ) = b f(x)dx. O isqurismìc utìc eðni ˆmesh sunèpei tou ex c genikìterou jewr mtoc. Je rhm 5.. (je rhm mèshc tim c tou oloklhrwtikoô logismoô). 'Estw f : [, b] R suneq c sunˆrthsh ki èstw g : [, b] R oloklhr simh sunˆrthsh me mh rnhtikèc timèc. Upˆrqei ξ [, b] ste (5..3) f(x)g(x)dx = f(ξ) g(x)dx.

8 To jemeli des je rhm tou ApeirostikoÔ LogismoÔ Apìdeixh. Oi f ki g eðni oloklhr simec, ˆr h f g eðni oloklhr simh sto [, b]. H f eðni suneq c sto [, b], ˆr pðrnei elˆqisth ki mègisth tim. 'Estw (5..4) m = min{f(x) : x b} ki M = mx{f(x) : x b}. AfoÔ h g pðrnei mh rnhtikèc timèc, èqoume (5..5) mg(x) f(x)g(x) Mg(x) gi kˆje x [, b]. Sunep c, (5..6) m g(x)dx f(x)g(x)dx M g(x)dx. AfoÔ g sto [, b], èqoume g(x)dx. DikrÐnoume dôo peript seic: n b g(x)dx =, tìte pì thn (5..6) blèpoume ìti f(x)g(x)dx =. 'Ar, h (5..3) isqôei gi kˆje ξ [, b]. Upojètoume loipìn ìti g(x)dx >. Tìte, pì thn (5..6) sumperðnoume ìti (5..7) m f(x)g(x)dx b g(x)dx M. AfoÔ h f eðni suneq c, to Je rhm Endiˆmeshc Tim c deðqnei ìti upˆrqei ξ [, b] ste (5..8) f(ξ) = f(x)g(x)dx b g(x)dx. 'Epeti to sumpèrsm. Pìrism 5... 'Estw f : [, b] R suneq c sunˆrthsh. Upˆrqei ξ [, b] ste (5..9) f(x)dx = f(ξ)(b ). Apìdeixh. 'Amesh sunèpei tou Jewr mtoc 5.., n jewr soume thn g : [, b] R me g(x) = gi kˆje x [, b]. Sthn epìmenh prˆgrfo j deðxoume (xnˆ) to Pìrism 5.., ut th forˆ sn ˆmesh sunèpei tou pr tou jemeli douc jewr mtoc tou ApeirostikoÔ LogismoÔ. 5. T jemeli dh jewr mt tou ApeirostikoÔ LogismoÔ Orismìc 5.. (ìristo olokl rwm). 'Estw f : [, b] R oloklhr simh sunˆrthsh. EÐdme ìti h f eðni oloklhr simh sto [, x] gi kˆje x [, b]. To ìristo olokl rwm thc f eðni h sunˆrthsh F : [, b] R pou orðzeti pì thn (5..) F (x) = x f(t)dt. Qrhsimopoi ntc to gegonìc ìti kˆje Riemnn oloklhr simh sunˆrthsh eðni frgmènh, j deðxoume ìti to ìristo olokl rwm mic oloklhr simhc sunˆrthshc eðni pˆntote suneq c sunˆrthsh.

5. T jemeli dh jewr mt tou ApeirostikoÔ LogismoÔ 83 Je rhm 5... 'Estw f : [, b] R oloklhr simh sunˆrthsh. olokl rwm F thc f eðni suneq c sunˆrthsh sto [, b]. To ìristo Apìdeixh. AfoÔ h f eðni oloklhr simh, eðni ex orismoô frgmènh. Dhld, upˆrqei M > ste f(x) M gi kˆje x [, b]. 'Estw x < y sto [, b]. Tìte, F (x) F (y) = y y x f(t)dt x f(t) dt M x y. y f(t)dt = f(t)dt x 'Ar, h F eðni Lipschitz suneq c (me stjerˆ M). MporoÔme n deðxoume kˆti isqurìtero: st shmeð sunèqeic thc f, h F eðni prgwgðsimh. Je rhm 5..3. 'Estw f : [, b] R oloklhr simh sunˆrthsh. suneq c sto x [, b], tìte h F eðni prgwgðsimh sto x ki An h f eðni (5..) F (x ) = f(x ). Apìdeixh. Upojètoume ìti < x < b (oi dôo peript seic x = x = b elègqonti ìmoi, me th sômbsh pou kˆnme sthn rq tou KeflÐou). Jètoume δ = min{x, b x }. An h < δ, tìte F (x + h) F (x) h ( f(x ) = x+h ) x f(t)dt f(t)dt f(x ) h ( = x+h ) x+h f(t)dt f(x )dt h = h x x+h x x [f(t) f(x )]dt. 'Estw ε >. H f eðni suneq c sto x, ˆr upˆrqei < δ < δ ste n x x < δ tìte f(x) f(x ) < ε. 'Estw < h < δ. () An < h < δ, tìte F (x + h) F (x) h f(x ) = h h h x+h x x+h x x+h [f(t) f(x )]dt f(t) f(x ) dt εdt = hε = ε. x h

84 To jemeli des je rhm tou ApeirostikoÔ LogismoÔ (b) An δ < h <, tìte F (x + h) F (x) f(x ) h = 'Epeti ìti dhld F (x ) = f(x ). h h h x x +h x x +h x x +h F (x + h) F (x ) lim = f(x ), h h [f(t) f(x )]dt f(t) f(x ) dt εdt = ( h)ε = ε. h 'Amesh sunèpei eðni to pr to jemeli dec je rhm tou ApeirostikoÔ LogismoÔ. Je rhm 5..4 (pr to jemeli dec je rhm tou ApeirostikoÔ LogismoÔ). An h f : [, b] R eðni suneq c, tìte to ìristo olokl rwm F thc f eðni prgwgðsimh sunˆrthsh ki (5..3) F (x) = f(x) gi kˆje x [, b]. Pìrism 5..5. 'Estw f : [, b] R suneq c sunˆrthsh. Upˆrqei ξ [, b] ste (5..4) f(x)dx = f(ξ)(b ). Apìdeixh. Efrmìzoume to je rhm mèshc tim c tou diforikoô logismoô gi th sunˆrthsh F (x) = x f(t) dt sto [, b]. Ac upojèsoume t r ìti f : [, b] R eðni mi suneq c sunˆrthsh. Mi prgwgðsimh sunˆrthsh G : [, b] R lègeti prˆgous thc f ( ntiprˆgwgoc thc f) n G (x) = f(x) gi kˆje x [, b]. SÔmfwn me to Je rhm 5..4, h sunˆrthsh F (x) = x f(t)dt eðni prˆgous thc f. An G eðni mi ˆllh prˆgous thc f, tìte G (x) F (x) = f(x) f(x) = gi kˆje x [, b], ˆr h G F eðni stjer sto [, b] (pl sunèpei tou jewr mtoc mèshc tim c). Dhld, upˆrqei c R ste (5..5) G(x) F (x) = c gi kˆje x [, b]. AfoÔ F () =, pðrnoume c = G(). Dhld, (5..6) lli c x f(t)dt = G(x) G() (5..7) G(x) = G() + x gi kˆje x [, b]. 'Eqoume loipìn deðxei to ex c: f(t)dt

5. T jemeli dh jewr mt tou ApeirostikoÔ LogismoÔ 85 Je rhm 5..6. 'Estw f : [, b] R suneq c sunˆrthsh ki èstw F (x) = x f(t)dt to ìristo olokl rwm thc f. An G : [, b] R eðni mi prˆgous thc f, tìte (5..8) G(x) = F (x) + G() = gi kˆje x [, b]. Eidikìter, x f(t)dt + G() f(x)dx = G(b) G(). ShmeÐwsh: Den eðni swstì ìti gi kˆje prgwgðsimh sunˆrthsh G : [, b] R isqôei h isìtht (5..9) G(b) G() = G (x)dx. Gi prˆdeigm, n jewr soume th sunˆrthsh G : [, ] R me G() = ki G(x) = x sin x n < x, tìte h G eðni prgwgðsimh sto [, ] llˆ h G den eðni frgmènh sunˆrthsh (elègxte to) opìte den mporoôme n milˆme gi to olokl rwm G. An ìmwc h G : [, b] R eðni prgwgðsimh ki h G eðni oloklhr simh sto [, b], tìte h (5..9) isqôei. Autì eðni to deôtero jemeli dec je rhm tou ApeirostikoÔ LogismoÔ. Je rhm 5..7 (deôtero jemeli dec je rhm tou ApeirostikoÔ LogismoÔ). 'Estw G : [, b] R prgwgðsimh sunˆrthsh. An h G eðni oloklhr simh sto [, b] tìte (5..) G (x)dx = G(b) G(). Apìdeixh. 'Estw P = { = x < x < < x n = b} mi dimèrish tou [, b]. Efrmìzontc to Je rhm Mèshc Tim c sto [x k, x k+ ], k =,,..., n, brðskoume ξ k (x k, x k+ ) me thn idiìtht (5..) G(x k+ ) G(x k ) = G (ξ k )(x k+ x k ). An, gi kˆje k n, orðsoume (5..) m k = inf{g (x) : x k x x k+ } ki M k = sup{g (x) : x k x x k+ }, tìte (5..3) m k G (ξ k ) M k, ˆr n (5..4) L(G, P ) G (ξ k )(x k+ x k ) U(G, P ).

86 To jemeli des je rhm tou ApeirostikoÔ LogismoÔ Dhld, n (5..5) L(G, P ) (G(x k+ ) G(x k )) = G(b) G() U(G, P ). AfoÔ h P tn tuqoôs ki h G eðni oloklhr simh sto [, b], pðrnontc supremum wc proc P sthn rister nisìtht ki infimum wc proc P sthn dexiˆ nisìtht thc (5..5), sumperðnoume ìti (5..6) pou eðni to zhtoômeno. G (x)dx G(b) G() 5.3 Mèjodoi olokl rwshc G (x)dx, T jewr mt ut c thc prgrˆfou {perigrˆfoun} dôo qr simec mejìdouc olokl rwshc: thn olokl rwsh ktˆ mèrh ki thn olokl rwsh me ntiktˆstsh. Sumbolismìc. An F : [, b] R, tìte sumfwnoôme n grˆfoume (5.3.) [F (x)] b = F (x) b := F (b) F (). Je rhm 5.3. (olokl rwsh ktˆ mèrh). 'Estw f, g : [, b] R prgwgðsimec sunrt seic. An oi f ki g eðni oloklhr simec, tìte (5.3.) Eidikìter, (5.3.3) x fg = (fg)(x) (fg)() f(x)g (x)dx = [f(x)g(x)] b Apìdeixh. H f g eðni prgwgðsimh ki x (5.3.4) (f g) (x) = f(x)g (x) + f (x)g(x) f g. f (x)g(x)dx. sto [, b]. Apì thn upìjesh, oi sunrt seic fg, f g eðni oloklhr simec, ˆr ki h (f g) eðni oloklhr simh. Apì to deôtero jemeli dec je rhm tou ApeirostikoÔ LogismoÔ, gi kˆje x [, b] èqoume (5.3.5) x fg + x f g = x O deôteroc isqurismìc prokôptei n jèsoume x = b. (fg) = (fg)(x) (fg)(). Mi efrmog eðni to {deôtero je rhm mèshc tim c tou oloklhrwtikoô logismoô}. Pìrism 5.3.. 'Estw f, g : [, b] R. Upojètoume ìti h f eðni suneq c sto [, b] ki h g eðni monìtonh ki suneq c prgwgðsimh sto [, b]. Tìte, upˆrqei ξ [, b] ste (5.3.6) f(x)g(x)dx = g() ξ f(x)dx + g(b) ξ f(x)dx.

5.3 Mèjodoi olokl rwshs 87 Apìdeixh. JewroÔme to ìristo olokl rwm F (x) = x f(t)dt thc f sto [, b]. Tìte, to zhtoômeno pðrnei thn ex c morf : upˆrqei ξ [, b] ste (5.3.7) F (x)g(x)dx = g()f (ξ) + g(b)(f (b) F (ξ)). H g eðni suneq c prgwgðsimh, ˆr mporoôme n efrmìsoume olokl rwsh ktˆ mèrh sto risterì mèloc. 'Eqoume (5.3.8) F (x)g(x)dx = F (b)g(b) F ()g() F (x)g (x)dx = F (b)g(b) F (x)g (x)dx, foô F () =. Efrmìzoume to je rhm mèshc tim c tou OloklhrwtikoÔ LogismoÔ: h g eðni monìtonh, ˆr h g dithreð prìshmo sto [, b]. H F eðni suneq c ki h g oloklhr simh, ˆr upˆrqei ξ [, b] ste (5.3.9) F (x)g (x)dx = F (ξ) Antikjist ntc sthn (5.3.8) pðrnoume (5.3.) g (x)dx = F (ξ)(g(b) g()). F (x)g(x) = F (b)g(b) F (ξ)(g(b) g()) = g()f (ξ) + g(b)(f (b) F (ξ)), dhld thn (5.3.7). Je rhm 5.3.3 (pr to je rhm ntiktˆstshc). 'Estw φ : [, b] R prgwgðsimh sunˆrthsh. Upojètoume ìti h φ eðni oloklhr simh. An I = φ([, b]) ki f : I R eðni mi suneq c sunˆrthsh, tìte (5.3.) f(φ(t))φ (t) dt = φ(b) φ() f(s) ds. Apìdeixh. H φ eðni suneq c, ˆr to I = φ([, b]) eðni kleistì diˆsthm. H f eðni suneq c sto I, ˆr eðni oloklhr simh sto I. OrÐzoume F : I R me (5.3.) F (x) = x φ() f(s) ds (prthr ste ìti to φ() den eðni prðtht ˆkro tou I, dhld h F den eðni prðtht to ìristo olokl rwm thc f sto I). AfoÔ h f eðni suneq c sto I, to pr to jemeli dec je rhm tou ApeirostikoÔ LogismoÔ deðqnei ìti h F eðni prgwgðsimh sto I ki F = f. 'Epeti ìti (5.3.3) PrthroÔme ìti f(φ(t))φ (t) dt = F (φ(t))φ (t) dt. (5.3.4) (F φ) φ = (F φ).

88 To jemeli des je rhm tou ApeirostikoÔ LogismoÔ H (F φ) φ eðni oloklhr simh sto [, b], ˆr h (F φ) eðni oloklhr simh sto [, b]. Apì to deôtero jemeli dec je rhm tou ApeirostikoÔ LogismoÔ pðrnoume (5.3.5) AfoÔ (f φ) φ = (F φ) φ = (F φ) = (F φ)(b) (F φ)(). (5.3.6) (F φ)(b) (F φ)() = φ(b) f φ() f = φ(b) φ() φ() φ() f, pðrnoume thn (5.3.). Je rhm 5.3.4 (deôtero je rhm ntiktˆstshc). 'Estw ψ : [, b] R suneq c prgwgðsimh sunˆrthsh, me ψ (x) gi kˆje x [, b]. An I = ψ([, b]) ki f : I R eðni mi suneq c sunˆrthsh, tìte (5.3.7) f(ψ(t)) dt = ψ(b) ψ() f(s)(ψ ) (s) ds. Apìdeixh. H ψ eðni suneq c ki den mhdenðzeti sto [, b], ˆr eðni pntoô jetik pntoô rnhtik sto [, b]. Sunep c, h ψ eðni gnhsðwc monìtonh sto [, b]. An, qwrðc periorismì thc genikìthtc, upojèsoume ìti h ψ eðni gnhsðwc Ôxous tìte orðzeti h ntðstrofh sunˆrthsh ψ : I R thc ψ sto I = ψ([, b]) = [ψ(), ψ(b)]. Efrmìzoume to pr to je rhm ntiktˆstshc gi thn f (ψ ) (prthr ste ìti h (ψ ) eðni suneq c sto I). 'Eqoume ψ(b) ψ() Autì podeiknôei thn (5.3.7). f (ψ ) = = = = 5.4 Genikeumèn oloklhr mt [(f (ψ ) ) ψ]ψ (f ψ) [(ψ ) ψ]ψ (f ψ) (ψ ψ) f ψ. Se ut n thn prˆgrfo epekteðnoume ton orismì tou oloklhr mtoc gi sunrt - seic pou den eðni frgmènec eðni orismènec se dist mt pou den eðni kleistˆ ki frgmèn. J rkestoôme se kˆpoiec bsikèc ki qr simec peript seic.. Upojètoume ìti b R b = + ki f : [, b) R eðni mi sunˆrthsh pou eðni oloklhr simh ktˆ Riemnn se kˆje diˆsthm thc morf c [, x], ìpou < x < b. An upˆrqei to (5.4.) lim x b x f(t) dt

5.4 Genikeumèn oloklhr mt 89 ki eðni prgmtikìc rijmìc, tìte lème ìti h f eðni oloklhr simh sto [, b) ki orðzoume f(t) dt = lim x b x f(t) dt. An to {ìrio} sthn (5.4.) eðni ± tìte lème ìti to f(t) dt poklðnei sto ±. Entel c nˆlog orðzeti to genikeumèno olokl rwm mic sunˆrthshc f : (, b] R (ìpou R = ) pou eðni oloklhr simh sto [x, b] gi kˆje < x < b, n eðni to n to teleutðo ìrio upˆrqei. f(t) dt = lim x + x f(t) dt, PrdeÐgmt () JewroÔme th sunˆrthsh f : [, ) R me f(x) = x. Gi kˆje x > èqoume Sunep c, x f(t) dt = lim x t dt = t x = x. x ( f(t) dt = lim ) =. x x (b) JewroÔme th sunˆrthsh f : [, ) R me f(x) = x. Gi kˆje x > èqoume Sunep c, x t dt = ln t x = ln x ln = ln x. f(t) dt = lim x x f(t) dt = lim ln x = +. x (g) JewroÔme th sunˆrthsh f : (, ] R me f(x) = ln x. Prthr ste ìti h f den eðni frgmènh: lim ln x =. Gi kˆje x (, ) èqoume x + Sunep c, x f(t) dt = lim x + ln t dt = t ln t t x= x ln x + x. x f(t) dt = lim x +( x ln x + x) =. (d) JewroÔme th sunˆrthsh f : [, ) R me f(x) = x. Prthr ste ìti h f den eðni frgmènh: lim x x = +. Gi kˆje x (, ) èqoume Sunep c, x t dt = t x = x. f(t) dt = lim x x f(t) dt = lim x ( x) =.

9 To jemeli des je rhm tou ApeirostikoÔ LogismoÔ (e) JewroÔme th sunˆrthsh f : [, ) R me f(x) = sin x. Gi kˆje x > èqoume x sin t dt = cos t x = cos x. AfoÔ to ìrio lim (cos x ) den upˆrqei, to genikeumèno olokl rwm sin t dt x den pðrnei kˆpoi tim.. Upojètoume ìti b R b = + ki R =. 'Estw f : (, b) R mi sunˆrthsh pou eðni oloklhr simh ktˆ Riemnn se kˆje kleistì diˆsthm [x, y], ìpou < x < y < b. JewroÔme tuqìn c (, b) ki exetˆzoume n upˆrqoun t genikeumèn oloklhr mt c f(t) dt ki c f(t) dt. An upˆrqoun ki t dôo, tìte lème ìti to genikeumèno olokl rwm f(t) dt upˆrqei ki eðni Ðso me f(t) dt = c f(t) dt + c f(t) dt. Prthr ste ìti, se ut n thn perðptwsh, h tim tou jroðsmtoc sto dexiì mèloc den exrtˆti pì thn epilog tou c sto (, b) (exhg ste gitð). Sunep c, to genikeumèno olokl rwm orðzeti klˆ me utìn ton trìpo. An kˆpoio pì t dôo genikeumèn oloklhr mt c f(t) dt ki f(t) dt den èqei tim, tìte lème ìti to c b f(t) dt den orðzeti (den èqei tim ). Stic peript seic pou kˆpoio pì t dôo ki t dôo genikeumèn oloklhr mt poklðnoun sto ± isqôoun t sun jh gi tic morfèc ±. PrdeÐgmt () JewroÔme th sunˆrthsh f : R R me f(x) = 'Omoi, x. 'Eqoume x + x t ln(x + ) f(t) dt = lim x t dt = lim = +. + x f(t) dt =. Sunep c, to f(t) dt den orðzeti: èqoume prosdiìristh morf (+ ) + ( ). (b) JewroÔme th sunˆrthsh f : R R me f(x) =. 'Eqoume x + 'Omoi, Sunep c, x f(t) dt = lim x t + dt = t dt = lim rctn x = π/. + x f(t) dt = π/. t + dt + t + dt = π + π = π.

5.4 Genikeumèn oloklhr mt 9 5.4þ To krit rio tou oloklhr mtoc 'Estw f : [, ) R + mh rnhtik sunˆrthsh, h opoð eðni oloklhr simh se kˆje diˆsthm [, x], ìpou x >. Se ut n thn perðptwsh, h sunˆrthsh F (x) := eðni Ôxous sto (, + ). Sunep c, to x f(t) dt = lim x f(t) dt x f(t) dt upˆrqei n ki mìno n h F eðni ˆnw frgmènh. Diforetikˆ, f(t) dt = +. AntÐstoiqo potèlesm eðqme deð gi thn sôgklish seir n k me mhrnhtikoôc ìrouc. Mi tètoi seirˆ sugklðnei n ki mìno n h koloujð (s n ) twn merik n jroismˆtwn thc eðni ˆnw frgmènh. Diforetikˆ, poklðnei sto +. To epìmeno je rhm dðnei èn krit rio sôgklishc gi seirèc pou grˆfonti sth morf f(k), ìpou f : [, + ) R eðni mi fjðnous mh-rnhtik sunˆrthsh. Je rhm 5.4.. 'Estw f : [, + ) R fjðnous sunˆrthsh me mh rnhtikèc timèc. JewroÔme thn koloujð ( k ) me k = f(k), k =,,.... Tìte, h seirˆ mh rnhtik n ìrwn k sugklðnei n ki mìno n to genikeumèno olokl rwm f(t) dt upˆrqei. Apìdeixh. Apì to gegonìc ìti h f eðni fjðnous prokôptei ˆmes ìti h f eðni oloklhr simh se kˆje diˆsthm [k, k + ] ki k+ = f(k + ) k+ k f(t) dt f(k) = k gi kˆje k N. An upojèsoume ìti h seirˆ k sugklðnei, tìte gi kˆje x > èqoume x 'Epeti ìti to f(t) dt [x]+ f(t) dt = [x] k+ k f(t) dt = lim x x [x] f(t) dt k k. f(t) dt upˆrqei. AntÐstrof, n to f(t) dt upˆrqei, gi kˆje n N èqoume n s n = f() + f() + + f(n) f() + = f() + n f(t) dt f() + f(t) dt. k+ k f(t) dt AfoÔ h koloujð (s n ) twn merik n jroismˆtwn thc k eðni ˆnw frgmènh, h seirˆ sugklðnei.. PrdeÐgmt

9 To jemeli des je rhm tou ApeirostikoÔ LogismoÔ () H seirˆ k= k ln k poklðnei diìti x ln x dy dt = lim dt = lim t ln t x t ln t x ln y = lim (ln(ln x) ln(ln )) = +. x (b) H seirˆ k= k(ln k) sugklðnei diìti x ln x ( dy dt = lim dt = lim t(ln t) x t(ln t) x ln y = lim x ln ) = ln x ln. 5.5 Ask seic Omˆd A'. 'Estw f : [, b] R oloklhr simh sunˆrthsh. DeÐxte ìti upˆrqei s [, b] ste s f(t)dt = s f(t)dt. MporoÔme pˆnt n epilègoume èn tètoio s sto noiktì diˆsthm (, b)?. 'Estw f : [, ] R oloklhr simh ki jetik sunˆrthsh ste f(x)dx =. DeÐxte ìti gi kˆje n N upˆrqei dimèrish { = t < t < < t n = } ste tk+ t k f(x)dx = n gi kˆje k =,,..., n. 3. 'Estw f : [, ] R suneq c sunˆrthsh. DeÐxte ìti upˆrqei s [, ] ste f(x)x dx = f(s) 3. 4. Upojètoume ìti h f : [, ] R eðni suneq c ki ìti x f(t)dt = x f(t)dt gi kˆje x [, ]. DeÐxte ìti f(x) = gi kˆje x [, ]. 5. 'Estw f, h : [, + ) [, + ). Upojètoume ìti h h eðni suneq c ki h f eðni prgwgðsimh. OrÐzoume DeÐxte ìti F (x) = h(f(x)) f (x). F (x) = f(x) h(t)dt. 6. 'Estw f : R R suneq c ki èstw δ >. OrÐzoume g(x) = x+δ x δ f(t)dt. DeÐxte ìti h g eðni prgwgðsimh ki breðte thn g.

5.5 Ask seis 93 7. 'Estw g, h : R R prgwgðsimec sunrt seic. OrÐzoume G(x) = g(x) h(x) t dt. DeÐxte ìti h G eðni prgwgðsimh sto R ki breðte thn G. 8. 'Estw f : [, + ) R suneq c sunˆrthsh. OrÐzoume F (x) = BreÐte thn F. x ( x ) f dt. t 9. 'Estw f : [, ] R suneq c. DeÐxte ìti, gi kˆje x [, ], x x ( u ) f(u)(x u)du = f(t)dt du.. 'Estw, b R me < b ki f : [, b] R suneq c prgwgðsimh sunˆrthsh. An P = { = x < x < < x n = b} eðni dimèrish tou [, b], deðxte ìti n f(x k+ ) f(x k ) f (x) dx.. 'Estw f : [, + ) [, + ) gnhsðwc Ôxous, suneq c prgwgðsimh sunˆrthsh me f() =. DeÐxte ìti, gi kˆje x >, x f(t) dt + f(x) f (t) dt = xf(x). Omˆd B'. 'Estw f : [, ] R suneq c prgwgðsimh sunˆrthsh me f() =. DeÐxte ìti gi kˆje x [, ] isqôei ( / f(x) f (t) dt). 3. 'Estw f : [, + ) R suneq c sunˆrthsh me f(x) gi kˆje x >, h opoð iknopoieð thn f(x) = x f(t)dt gi kˆje x. DeÐxte ìti f(x) = x gi kˆje x. 4. 'Estw f : [, b] R suneq c prgwgðsimh sunˆrthsh. DeÐxte ìti lim n f(x) cos(nx)dx = ki lim n f(x) sin(nx)dx =.

94 To jemeli des je rhm tou ApeirostikoÔ LogismoÔ 5. Exetˆste wc proc th sôgklish tic koloujðec n = π sin(nx)dx ki b n = π sin(nx) dx. 6. 'Estw f : [, + ) R suneq c prgwgðsimh sunˆrthsh. DeÐxte ìti upˆrqoun suneqeðc, Ôxousec ki jetikèc sunrt seic g, h : [, + ) R ste f = g h.

Kefˆlio 6 Teqnikèc olokl rwshc Se utì to Kefˆlio perigrˆfoume, qwrðc idiðterh usthrìtht, tic bsikèc mejìdouc upologismoô oloklhrwmˆtwn. DÐneti mi sunˆrthsh f ki jèloume n broôme mi ntiprˆgwgo thc f, dhld mi sunˆrthsh F me thn idiìtht F = f. Tìte, f(x)dx = F (x) + c. 6. Olokl rwsh me ntiktˆstsh 6.þ PÐnkc stoiqeiwd n oloklhrwmˆtwn Kˆje tôpoc prg gishc F (x) = f(x) mc dðnei ènn tôpo olokl rwshc: h F eðni ntiprˆgwgoc thc f. MporoÔme ètsi n dhmiourg soume ènn pðnk bsik n oloklhrwmˆtwn, ntistrèfontc touc tôpouc prg gishc twn pio bsik n sunrt - sewn: x dx = x+ +,, dx = ln x + c x e x dx = e x + c, sin x dx = cos x + c cos x dx = sin x + c, cos dx = tn x + c x sin dx = cot x + c, dx = rcsin x + c x x dx = rctn x + c. + x 6.bþ Upologismìc tou f(φ(x))φ (x) dx H ntiktˆstsh u = φ(x), du = φ (x) dx mc dðnei f(φ(x))φ (x) dx = f(u) du, u = φ(x). An to olokl rwm dexiˆ upologðzeti eukolìter, jètontc ìpou u thn φ(x) upologðzoume to olokl rwm risterˆ.

96 Teqnikès olokl rwshs PrdeÐgmt () Gi ton upologismì tou jètoume u = rctn x. Tìte, du = 'Epeti ìti (b) Gi ton upologismì tou rctn x + x dx dx +x ki ngìmste sto u du = u + c. rctn x (rctn x) dx = + c. + x tn x dx = sin x cos x dx jètoume u = cos x. Tìte, du = sin x dx ki ngìmste sto du = ln u + c. u 'Epeti ìti tn x dx = ln cos x + c. (g) Gi ton upologismì tou cos( x) x dx jètoume u = x. Tìte, du = dx ki ngìmste sto x cos u du = sin u + c. 'Epeti ìti cos( x) x dx = sin( x) + c. 6.gþ Trigwnometrikˆ oloklhr mt Oloklhr mt pou perièqoun dunˆmeic ginìmen trigwnometrik n sunrt sewn m- poroôn n nqjoôn se ploôster n qrhsimopoi soume tic bsikèc trigwnometrikèc tutìthtec: sin x + cos x =, + tn x = cos x + cot x = sin x, + cos x cos x = sin cos x x =, sin x = sin x cos x cos( b)x cos( + b)x sin( + b)x + sin( b)x sin x sin bx =, sin x cos bx = cos( + b)x + cos( b)x cos x cos bx =.

6. Olokl rwsh me ntiktˆstsh 97 PrdeÐgmt () Gi ton upologismì tou cos x dx qrhsimopoioôme thn cos +cos x x = : èqoume + cos x cos x dx = dx = dx + cos x dx = x + sin x 4 Me ton Ðdio trìpo mporoôme n uplogðsoume to cos 4 x dx, qrhsimopoi ntc thn ( ) + cos x cos 4 x = = cos x + + cos x = cos x + cos 4x + +. 4 4 4 8 + c. (b) Gi ton upologismì tou sin 5 x dx = sin 4 x sin x dx = ( cos x) sin x dx eðni protimìterh h ntiktˆstsh u = cos x. Tìte, du = sin x dx ki ngìmste sto ( u ) du = u + u3 3 u5 5 + c. 'Epeti ìti sin 5 x dx = cos x + cos3 x cos5 x + c. 3 5 Thn Ðdi mèjodo mporoôme n qrhsimopoi soume gi opoiod pote olokl rwm thc morf c cos m x sin n x dx n ènc pì touc ekjètec m, n eðni perittìc ki o ˆlloc ˆrtioc. Gi prˆdeigm, n m = 3 ki n = 4, grˆfoume cos 3 x sin 4 x dx = ( sin x) sin 4 x cos x dx ki, me thn ntiktˆstsh u = sin x, ngìmste sto plì olokl rwm ( u )u 4 du. (g) DÔo qr sim oloklhr mt eðni t tn x dx ki cot x dx. Gi to pr to grˆfoume ( ) tn x dx = cos x dx = [(tn x) ] dx = tn x x + c, ki, ìmoi, gi to deôtero grˆfoume ( ) cot x dx = sin x dx = [( cot x) ] dx = cot x x + c.

98 Teqnikès olokl rwshs 6.dþ Upologismìc tou f(x) dx me thn ntiktˆstsh x = φ(t) H ntiktˆstsh x = φ(t), dx = φ (t) dt ìpou φ ntistrèyimh sunˆrthsh mc dðnei f(x) dx = f(φ(t))φ (t) dt. An to olokl rwm dexiˆ upologðzeti eukolìter, jètontc ìpou t thn φ (x) upologðzoume to olokl rwm risterˆ. PrdeÐgmt: trigwnometrikèc ntiktstˆseic () Se oloklhr mt pou perièqoun thn x jètoume x = sin t. Tìte, x = cos t ki dx = cos t dt. Gi prˆdeigm, gi ton upologismì tou dx x 9 x, n jèsoume x = 3 sin t, tìte dx = 3 cos t dt ki 9 x = 3 cos t, ki ngìmste sto 3 cos t dt 9 sin t(3 cos t) = dt 9 sin t = cot t + c. 9 Tìte, pì thn pðrnoume cot t = cos t sin sin t = t 9 x =, sin t x dx 9 x x 9 x = + c. 9x (b) Se oloklhr mt pou perièqoun thn x jètoume x = / cos t. Tìte, Gi prˆdeigm, gi ton upologismì tou n jèsoume x = ngìmste sto cos t x = tn t ki dx = sin t cos t dt. x 4 dx, x sin t tn t, tìte dx = cos t dt = cos t dt ki x 4 = tn t, ki tn t tn t / cos t cos t dt = tn t dt = tn t t + c. AfoÔ t = rctn x 4, pðrnoume x 4 dx = x x x 4 4 rctn + c.

6. Olokl rwsh ktˆ mèrh 99 (g) Se oloklhr mt pou perièqoun thn x + jètoume x = tn t. Tìte, x + = ki dx = cos t cos t dt. Gi prˆdeigm, gi ton upologismì tou x + dx, x 4 n jèsoume x = tn t, tìte dx = cos t dt ki x + = cos t, ki ngìmste sto cos t cos t tn 4 t cos t dt = sin 4 t dt = 3 sin 3 t + c. AfoÔ t = rctn x, blèpoume ìti sin t = tn t cos t = x x + x + x 4 dx = (x + ) 3/ 3x 3 + c. 6. Olokl rwsh ktˆ mèrh O tôpoc thc olokl rwshc ktˆ mèrh eðni: f(x)g (x) dx = f(x)g(x) f (x)g(x) dx, ki telikˆ pðrnoume ki prokôptei ˆmes pì thn (fg) = fg + f g, n oloklhr soume t dôo mèlh thc. Suqnˆ, eðni eukolìtero n upologðsoume to olokl rwm sto dexiì mèloc. PrdeÐgmt () Gi ton upologismì tou x log x dx grˆfoume x log x dx = (x ) log x dx = x log x x dx = x log x x 4 + c. (b) Gi ton upologismì tou x cos x dx grˆfoume x cos x dx = x(sin x) dx = x sin x sin x dx = x sin x + cos x + c. (g) Gi ton upologismì tou e x sin x dx grˆfoume I = e x sin x dx = (e x ) sin x dx = e x sin x e x cos x dx = e x sin x (e x ) cos x dx = e x sin x e x cos x + e x (cos x) dx = e x (sin x cos x) e x sin x dx = e x (sin x cos x) I. 'Epeti ìti e x sin x dx = ex (sin x cos x) + c.

Teqnikès olokl rwshs (d) Gi ton upologismì tou x sin x dx qrhsimopoi ntc thn tutìtht sin x = cos(x) grˆfoume x sin x dx = x dx x cos(x) Gi to deôtero olokl rwm, qrhsimopoioôme thn ntiktˆstsh u = x ki olokl rwsh ktˆ mèrh ìpwc sto (b). (e) Gi ton upologismì tou log(x + x) dx grˆfoume log(x+ x) dx = (x) log(x+ x) dx = x log(x+ x) Ktìpin, efrmìzoume thn ntiktˆstsh u = x. 6.3 Olokl rwsh rht n sunrt sewn dx. ( x x + + ) x dx. x Se ut thn prˆgrfo perigrˆfoume mi mèjodo me thn opoð mporeð kneðc n upologðsei to ìristo olokl rwm opoisd pote rht c sunˆrthshc (6.3.) f(x) = p(x) q(x) = nx n + n x n + + x + b m x m + b m x m + + b x + b. H pr th prt rhsh eðni ìti mporoôme pˆnt n upojètoume ìti n < m. An o bjmìc n tou rijmht p(x) eðni meglôteroc Ðsoc pì ton bjmì m tou pronomst q(x), tìte diiroôme to p(x) me to q(x): upˆrqoun polu num π(x) ki υ(x) ste o bjmìc tou υ(x) n eðni mikrìteroc pì m ki (6.3.) p(x) = π(x)q(x) + υ(x). Tìte, (6.3.3) f(x) = π(x)q(x) + υ(x) q(x) = π(x) + υ(x) q(x). Sunep c, gi ton upologismì tou f(x) dx mporoôme t r n upologðsoume qwristˆ to π(x) dx (plì olokl rwm poluwnumik c sunˆrthshc) ki to υ(x) q(x) dx (rht sunˆrthsh me thn prìsjeth idiìtht ìti deg(υ) < deg(q)). Upojètoume loipìn sth sunèqei ìti f = p/q ki deg(p) < deg(q). MporoÔme epðshc n upojèsoume ìti n = b m =. QrhsimopoioÔme t r to gegonìc ìti kˆje polu numo nlôeti se ginìmeno prwtobˆjmiwn ki deuterobˆjmiwn ìrwn. To q(x) = x m + + b x + b grˆfeti sth morf (6.3.4) q(x) = (x α ) r (x α k ) r k (x + β x + γ ) s (x + β l x + γ l ) s l. Oi α,..., α k eðni oi prgmtikèc rðzec tou q(x) (ki r j eðni h pollplìtht thc rðzc α j ) en oi ìroi x + β i x + γ i eðni t ginìmen (x z i )(x z i ) ìpou z i oi migdikèc rðzec tou q(x) (ki s i eðni h pollplìtht thc rðzc z i ). Prthr ste ìti kˆje ìroc thc morf c x + β i x + γ i èqei rnhtik dikrðnous. EpÐshc, oi k, s ki r + + r k + s + + s l = m (o bjmìc tou q(x)).

6.3 Olokl rwsh rht n sunrt sewn Grˆfoume thn f(x) sth morf x n + n x n + + x + (6.3.5) f(x) = (x α ) r (x α k ) r k (x + β x + γ ) s (x + β l x + γ l ) s, l ki thn {nlôoume se plˆ klˆsmt}: upˆrqoun suntelestèc A jt, B it, Γ it ste f(x) = A + x α + + A k x α k + A (x α ) + + A r (x α ) r A k (x α k ) + + A kr (x α k ) r k + B x + Γ x + B x + Γ + β x + γ (x + β x + γ ) + + B s x + Γ s (x + β x + γ ) s + + B lx + Γ l x + B lx + Γ l + β l x + γ l (x + β l x + γ l ) + + B ls x + Γ lsl (x + β l x + γ l ) s. l H eôresh twn suntelest n gðneti wc ex c: pollplsiˆzoume t dôo mèlh thc isìthtc me to q(x) (prthr ste ìti isoôti me to elˆqisto koinì pollplˆsio twn pronomst n tou dexioô mèlouc). ProkÔptei tìte mi isìtht poluwnômwn. Exis nontc touc suntelestèc touc, pðrnoume èn sôsthm m exis sewn me m gn stouc: touc A j,..., A jrj, B i,..., B isi, Γ i,..., Γ isi, j =,..., k, i =,..., l. Metˆ pì utì to b m, qrhsimopoi ntc thn grmmikìtht tou oloklhr mtoc, ngìmste ston upologismì oloklhrwmˆtwn twn ex c dôo morf n: () Oloklhr mt thc morf c dx. Autˆ upologðzonti ˆmes: n (x α) k k tìte (6.3.6) ki n k = tìte (6.3.7) (x α) k dx = + c, (k )(x α) k dx = ln x α + c. x α (b) Oloklhr mt thc morf c Bx+Γ dx, ìpou to x + bx + γ èqei rnhtik dikrðnous. Grˆfontc Bx + Γ = B (x + b) + ( ) (x +bx+γ) k Γ Bb, ngìmste st oloklhr mt x + b (6.3.8) (x + bx + γ) k dx ki (x + bx + γ) k dx. To pr to upologðzeti me thn ntiktˆstsh y = x + bx + γ (exhg ste gitð). Gi to deôtero, grˆfoume pr t x +bx+γ = ( ) + x + b 4γ b 4 ki me thn ntiktˆstsh x + b = 4γ b y ngìmste (exhg ste gitð) ston upologismì oloklhrwmˆtwn thc morf c (6.3.9) I k = (y + ) k dy.

Teqnikès olokl rwshs O upologismìc tou I k bsðzeti sthn ndromik sqèsh (6.3.) I k+ = y k (y + ) k + k k I k. Gi thn pìdeixh thc (6.3.) qrhsimopoioôme olokl rwsh ktˆ mèrh. Grˆfoume I k = dx (y + ) k = = = = (y) (y + ) k dy = y (y + ) k + k y y (y + ) k + k + (y dy + ) k+ y (y + ) k + k y (y + ) k + ki k ki k+. (y dy k + ) k 'Epeti to zhtoômeno. GnwrÐzoume ìti (6.3.) I = y dy = rctn y + c, + y (y dy + ) k+ (y dy + ) k+ ˆr, qrhsimopoi ntc thn (6.3.), mporoôme didoqikˆ n broôme t I, I 3,.... PrdeÐgmt () Gi ton upologismì tou oloklhr mtoc zhtˆme, b, c R ste Grˆfoume 3x + 6 x 3 + x x dx = 3x + 6 x(x )(x + ) = x + 3x + 6 x(x )(x + ) dx, b x + c x +. 3x + 6 x(x )(x + ) = (x )(x + ) + bx(x + ) + cx(x ) x(x )(x + ) = ( + b + c)x + ( + b c)x, x(x )(x + ) ki lônoume to sôsthm + b + c = 3, + b c =, = 6. H lôsh eðni: = 3, b = 3 ki c = 3. Sunep c, 3x + 6 dx dx = 3 x(x )(x + ) x + 3 dx x + 3 dx x + = 3 ln x + 3 ln x + 3 ln x + + c.

6.3 Olokl rwsh rht n sunrt sewn 3 (b) Gi ton upologismì tou oloklhr mtoc 5x + x + x 3 + 3x 4 dx = 5x + x + (x )(x + ) dx, zhtˆme, b, c R ste Grˆfoume 5x + x + (x )(x + ) = x + b x + + c (x + ). 5x + x + (x )(x + ) = (x + ) + b(x )(x + ) + c(x ) (x )(x + ) ki lônoume to sôsthm = ( + b)x + (4 + b + c)x + (4 b c) (x )(x + ), + b = 5, 4 + b + c =, 4 b c =. H lôsh eðni: =, b = 3 ki c =. Sunep c, 5x + x + (x )(x + ) dx = dx x + 3 (g) Gi ton upologismì tou oloklhr mtoc x + x 5 x 4 + x 3 x + x dx = zhtˆme, b, c, d, e R ste Ktl goume sthn dx x + + dx (x + ) = ln x + 3 ln x + x + + c. x + (x )(x + ) dx, x + (x )(x + ) = x + bx + c x + + dx + e (x + ). x + = (x + ) + (bx + c)(x )(x + ) + (dx + e)(x ) ki lônoume to sôsthm + b =, b + c =, + b c + d =, b + c d + e =, c e =. H lôsh eðni: = /, b = /, c = /, d = ki e =. Sunep c, x + x 5 x 4 + x 3 x + x dx = dx x x + x + dx x (x + ) dx = dx x x 4 x + dx dx x + (x + ) (x + ) dx = ln x 4 ln x + rctn x + x + + c.

4 Teqnikès olokl rwshs 6.4 Kˆpoiec qr simec ntiktstˆseic 6.4þ Rhtèc sunrt seic twn cos x ki sin x Gi ton upologismì oloklhrwmˆtwn thc morf c R(cos x, sin x) dx ìpou R(u, v) eðni phlðko poluwnômwn me metblhtèc u ki v, suqnˆ qrhsimopoioôme thn ntiktˆstsh Prthr ste ìti u = tn x. ki cos x = cos x sin x cos x + sin x = tn x + tn x = u + u sin x = sin x cos x = tn x cos x = tn x + tn x = u + u. EpÐshc, du dx = cos x = +tn x, dhld 'Etsi, ngìmste sto olokl rwm dx = du + u. ( ) u R + u, u + u + u du. ( ) u Dedomènou ìti h sunˆrthsh F (u) = R u +u, +u +u eðni rht sunˆrthsh tou u, to teleutðo olokl rwm upologðzeti me th mèjodo pou perigrˆyme sthn Prˆgrfo 6.3. PrdeÐgmt. () Gi ton upologismì tou oloklhr mtoc + sin x cos x dx jètoume u = tn x sto olokl rwm. AfoÔ dx = +u du, cos x = u +u ki sin x = u +u, ngìmste ( + u) u ( + u ) du, to opoðo upologðzeti me nˆlush se plˆ klˆsmt. () Gi ton upologismì tou oloklhr mtoc x + sin x dx

6.4 Kˆpoies qr simes ntiktstˆseis 5. AfoÔ dx = +u du ki sin x = u +u, ngìmste sto olok- jètoume u = tn x l rwm rctn u + u +u + u du = 4 = 4 rctn u ( + u) du ( rctn u ) du + u = 4 rctn u + u + 4 ( + u )( + u) du. To teleutðo olokl rwm upologðzeti me nˆlush se plˆ klˆsmt. 6.4bþ Oloklhr mt lgebrik n sunrt sewn eidik c morf c Perigrˆfoume ed kˆpoiec ntiktstˆseic pou qrhsimopoioônti gi ton upologismì oloklhrwmˆtwn thc morf c R(x, x ) dx, R(x, x ) dx, R(x, x + ) dx, ìpou R(u, v) eðni phlðko poluwnômwn me metblhtèc u ki v. () Gi to olokl rwm R(x, x ) dx, kˆnoume pr t thn llg metblht c x = sin t. AfoÔ x = cos t ki dx = cos t dt, ngìmste sto olokl rwm R(sin t, cos t) cos t dt, to opoðo upologðzeti me thn ntiktˆstsh thc prohgoômenhc upoprgrˆfou (rht sunˆrthsh twn cos t ki sin t). (b) Gi to olokl rwm R(x, x ) dx, mi idè eðni n qrhsimopoi soume thn llg metblht c x =. Tìte, cos t x = sin t cos t ètsi sto olokl rwm ( R sin t, sin t ) sin t cos t cos t dt = ki dx = sin t cos t R (cos t, sin t) dt dt. Angìmste gi kˆpoi rht sunˆrthsh R (u, v), to opoðo upologðzeti me thn ntiktˆstsh thc prohgoômenhc upoprgrˆfou (rht sunˆrthsh twn cos t ki sin t). EÐni ìmwc protimìtero n qrhsimopoi soume thn ex c llg metblht c: Tìte, u = x + x. x = u + u, x = u u, dx = u u du. Angìmste ètsi sto rhtì olokl rwm ( u ) + R u, u u u u du to opoðo upologðzeti me nˆlush se plˆ klˆsmt.

6 Teqnikès olokl rwshs (b) Gi to olokl rwm R(x, x + ) dx, mi idè eðni n qrhsimopoi soume thn llg metblht c x = cot t. Tìte, x = sin t ki dx = dt. Angìmste ètsi sto olokl rwm sin t ( R cos t ) sin t, sin t sin t dt = R (cos t, sin t) dt gi kˆpoi rht sunˆrthsh R (u, v), to opoðo upologðzeti me thn ntiktˆstsh thc prohgoômenhc upoprgrˆfou (rht sunˆrthsh twn cos t ki sin t). EÐni ìmwc protimìtero n qrhsimopoi soume thn ex c llg metblht c: Tìte, u = x + x +. x = u u, x = u + u, dx = u + u du. Angìmste ètsi sto rhtì olokl rwm ( u ) R u, u + u + u u du to opoðo upologðzeti me nˆlush se plˆ klˆsmt. PrdeÐgmt () Gi ton upologismì tou oloklhr mtoc x dx jètoume x = (x u). IsodÔnm, x = u +. Tìte, u dx = u u du ki x u = u u, opìte ngìmste ston upologismì tou (u ) 4u 3 du. (b) Gi ton upologismì tou oloklhr mtoc x x + dx jètoume u = x + x +. Tìte, x = u u, x = u + u, dx = u + u du. Angìmste ètsi sto olokl rwm u du to opoðo upologðzeti me nˆlush se plˆ klˆsmt.

6.5 Ask seic Omˆd A'. UpologÐste t kìlouj oloklhr mt: x x + x + dx, x + x + (x + 3)(x ) dx, 6.5 Ask seis 7 3x + 3x + x 3 + x + x + dx.. UpologÐste t kìlouj oloklhr mt: dx x 4 +, dx x + 3 x, dx x x, dx + e x. 3. UpologÐste t kìlouj oloklhr mt: cos 3 x dx, cos x sin 3 x dx, tn x dx, dx cos 4 x, tn x dx. 4. Qrhsimopoi ntc olokl rwsh ktˆ mèrh, deðxte ìti: gi kˆje n N, dx (x + ) n+ = x n (x + ) n + n dx n (x + ) n. 5. UpologÐste t kìlouj oloklhr mt: x (x 4)(x ) dx, x cos x dx, log(x + x) dx, x + sin x dx, ( + x)( + x ) dx, e x sin x dx, x x dx, cos 3 x sin x dx, x sin x dx x log x dx x + 4 (x + )(x ) dx dx (x + x + ). 6. UpologÐste t oloklhr mt sin(log x) dx, x log( x) dx. x 7. UpologÐste t oloklhr mt x rctn x ( + x ) dx, xe x ( + x) dx.

8 Teqnikès olokl rwshs 8. UpologÐste t oloklhr mt e x dx, + ex log(tn x) cos x dx. 9. UpologÐste t oloklhr mt 5 π 4 x cos x dx, ( ) x log + x dx, π 4 π 4 π 4 tn 3 x cos 3 x dx x tn x dx.. UpologÐste t kìlouj embdˆ: () Tou qwrðou pou brðsketi sto pr to tetrthmìrio ki frˆsseti pì tic grfikèc prstˆseic twn sunrt sewn f(x) = x, g(x) = x ki pì ton x-ˆxon. (b) Tou qwrðou pou frˆsseti pì tic grfikèc prstˆseic twn sunrt sewn f(x) = cos x ki g(x) = sin x sto diˆsthm [ π 4, 5π 4 ]. Omˆd B'. UpologÐste t oloklhr mt + sin x cos x dx, sin x dx, ( + x ) dx, x rctn x dx, x ( + x ) dx, x x + dx, x x dx x dx.. UpologÐste to olokl rwm π x sin x + cos x dx. 3. UpologÐste to olokl rwm π sin x sin x + cos x dx. 4. UpologÐste to olokl rwm π 4 log( + tn x) dx. 5. DeÐxte ìti to genikeumèno olokl rwm x p dx

6.5 Ask seis 9 den eðni pepersmèno gi knèn p R. 6. UpologÐste t kìlouj genikeumèn oloklhr mt: xe x dx, dx, log x dx. x 7. DeÐxte ìti, gi kˆje n N, e x x n dx = n! 8. BreÐte t ìri x lim x + x3 e x6 e t dt, 3 x lim x + x 4 e t sin t dt.

Kefˆlio 7 Je rhm Tylor 7. Je rhm Tylor Orismìc 7... 'Estw f : [, b] R ki èstw x [, b]. Upojètoume ìti h f eðni n forèc prgwgðsimh sto x. To polu numo Tylor tˆxhc n thc f sto x eðni to polu numo T n,f,x : R R pou orðzeti wc ex c: (7..) T n,f,x (x) = dhld, (7..) n T n,f,x (x) = f(x ) + f (x )(x x ) + f (x ) f (k) (x ) (x x ) k, k! (x x ) + + f (n) (x ) (x x ) n. n! To upìloipo Tylor tˆxhc n thc f sto x eðni h sunˆrthsh R n,f,x : [, b] R pou orðzeti wc ex c: (7..3) R n,f,x (x) = f(x) T n,f,x (x). 'Otn x =, sunhjðzoume n onomˆzoume t T n,f, ki R n,f, polu numo McLurin ki upìloipo McLurin thc f ntðstoiq. Prt rhsh 7... PrgwgÐzontc to T n,f,x blèpoume ìti: T n,f,x (x) = T n,f,x (x) = T n,f,x (x) = T (n) n,f,x (x) = n n k= n f (k) (x ) (x x ) k, ˆr T n,f,x (x ) = f(x ), k! f (k) (x ) (k )! (x x ) k, ˆr T n,f,x (x ) = f (x ), f (k) (x ) (k )! (x x ) k, ˆr T n,f,x (x ) = f (x ), n f (k) (x ) (k n)! (x x ) k n, ˆr T (n) n,f,x (x ) = f (n) (x ). k=n

Je rhm Tylor Dhld, to polu numo Tylor tˆxhc n thc f sto x iknopoieð tic (7..3) T (k) n,f,x (x ) = f (k) (x ), k =,,..., n ki eðni to mondikì polu numo bjmoô to polô Ðsou me n pou èqei ut thn idiìtht (exhg ste gitð). Prt rhsh 7..3. 'Estw f : [, b] R ki èstw x [, b]. Upojètoume ìti h f eðni n forèc prgwgðsimh sto [, b] ki n forèc prgwgðsimh sto x. Prthr ste ìti (7..4) T n,f,x (x) = ki (7..5) T n,f,x (x) = n n s= f (k) (x ) (k )! (x x ) k Jètontc k = s + sthn (7..5) sumperðnoume ìti (7..6) T n,f,x = T n,f,x. 'Epeti ìti f (s+) (x ) (x x ) s. s! (7..7) R n,f,x = R n,f,x. Prìtsh 7..4. 'Estw f : [, b] R ki èstw x [, b]. Upojètoume ìti h f eðni n forèc prgwgðsimh sto [, b] ki n forèc prgwgðsimh sto x. Tìte, (7..8) lim x x R n,f,x (x) (x x ) n =. Apìdeixh. Me epgwg wc proc n. Gi n = èqoume ˆr (7..9) R,f,x (x) = f(x) f(x ) f (x )(x x ), R,f,x (x) x x = f(x) f(x ) x x f (x ) ìtn x x, pì ton orismì thc prg gou sto shmeðo x. Upojètoume ìti h prìtsh isqôei gi n = m ki gi kˆje sunˆrthsh pou iknopoieð tic upojèseic. 'Estw f : [, b] R, m forèc prgwgðsimh sto [, b] ki m + forèc prgwgðsimh sto x. Tìte, (7..) lim x x R m+,f,x (x) = lim x x (x x ) m+ = ki (7..) lim x x R m+,f,x (x) [(x x ) m+ ] = lim x x R m,f,x (x) (m + )(x x ) m = pì thn epgwgik upìjesh gi thn f. Efrmìzontc ton knìn l Hospitl oloklhr noume to epgwgikì b m.

7. Je rhm Tylor 3 L mm 7..5. 'Estw p polu numo bjmoô to polô Ðsou me n to opoðo iknopoieð thn p(x) (7..) lim x x (x x ) n =. Tìte, p. Apìdeixh. Me epgwg wc proc n. Gi to epgwgikì b m prthroôme pr t ìti p(x) (7..3) p(x ) = lim p(x) = lim x x x x (x x ) n (x x ) n =, Sunep c, p(x ) =. 'Ar, (7..4) p(x) = (x x )p (x), ìpou p polu numo bjmoô to polô Ðsou me n to opoðo iknopoieð thn (7..5) lim x x p (x) = lim (x x ) n x x p(x) (x x ) n =. An upojèsoume ìti h Prìtsh isqôei gi ton n, tìte p ˆr p.. H Prìtsh 7..4 ki to L mm 7..5 podeiknôoun ton ex c qrkthrismì tou poluwnômou Tylor T n,f,x : Je rhm 7..6. 'Estw f : [, b] R ki èstw x [, b]. Upojètoume ìti h f eðni n forèc prgwgðsimh sto [, b] ki n forèc prgwgðsimh sto x. Tìte, to polu numo Tylor tˆxhc n thc f sto x eðni to mondikì polu numo T bjmoô to polô Ðsou me n to opoðo iknopoieð thn (7..6) lim x x f(x) T (x) (x x ) n =. Apìdeixh. H Prìtsh 7..4 deðqnei ìti to T n,f,x iknopoieð thn (7..6). Gi th mondikìtht rkeð n prthr sete ìti n dôo polu num T, T bjmoô to polô Ðsou me n iknopoioôn thn (7..6), tìte to polu numo p := T T iknopoieð thn (7..). Apì to L mm 7..5 sumperðnoume ìti T T. Prt rhsh 7..7. To Je rhm 7..6 mc dðnei ènn èmmeso trìpo gi n brðskoume to polu numo Tylor tˆxhc n mic sunˆrthshc f se kˆpoio shmeðo x. ArkeÐ n broôme èn polu numo bjmoô to polô Ðsou me n to opoðo iknopoieð thn (7..6). (i) H sunˆrthsh f(x) = x eðni ˆpeirec forèc prgwgðsimh sto (, ) ki èqoume dei ìti x = + x + x + + x n + gi kˆje x <. J deðxoume ìti, gi kˆje n, PrthroÔme ìti T n,f, (x) = T n (x) := + x + + x n. f(x) T n (x) = x xn+ = xn+ x x.

4 Je rhm Tylor 'Ar, f(x) T n (x) x lim x x n = lim x x =, ki to zhtoômeno prokôptei pì to Je rhm 7..6. (ii) H sunˆrthsh f(x) = +x eðni ˆpeirec forèc prgwgðsimh sto R ki èqoume dei ìti + x = x + x 4 + + ( ) n x n + gi kˆje x <. J deðxoume ìti, gi kˆje n, T n,f, (x) = T n+,f, (x) = T n (x) := x + x 4 + ( ) n x n. PrthroÔme ìti 'Ar, ki (profn c) f(x) T n (x) = + x ( )n+ x n+ + x = ( )n+ x n+ + x. f(x) T n (x) ( ) n+ x lim x x n+ = lim x + x =, f(x) T n (x) ( ) n+ x lim x x n = lim x + x =, opìte to zhtoômeno prokôptei pì to Je rhm 7..6. To Je rhm Tylor dðnei eôqrhstec ekfrˆseic gi to upìloipo Tylor R n,f,x tˆxhc n mic sunˆrthshc f se kˆpoio shmeðo x. Je rhm 7..8 (Je rhm Tylor). 'Estw f : [, b] R mi sunˆrthsh n + forèc prgwgðsimh sto [, b] ki èstw x [, b]. Tìte, gi kˆje x [, b], (i) Morf Cuchy tou upoloðpou Tylor: Upˆrqei ξ metxô twn x ki x ste (7..7) R n,f,x (x) = f (n+) (ξ) (x ξ) n (x x ). n! (ii) Morf Lgrnge tou upoloðpou Tylor: Upˆrqei ξ metxô twn x ki x ste (7..8) R n,f,x (x) = f (n+) (ξ) (n + )! (x x ) n+. (iii) Oloklhrwtik morf tou upoloðpou Tylor: An h f (n+) eðni oloklhr simh sunˆrthsh, tìte (7..9) R n,f,x (x) = n! x x f (n+) (t)(x t) n dt.

7. Je rhm Tylor 5 Apìdeixh. StjeropoioÔme to x [, b] ki orðzoume φ : [, b] R me n f (k) (t) (7..) φ(t) = R n,f,t (x) = f(x) (x t) k. k! PrgwgÐzontc wc proc t blèpoume ìti n ( f φ (t) = f (k+) (t) (t) (x t) k f (k) ) (t) (x t)k k! (k )! = f (n+) (t) (x t) n, n! foô to mesðo ˆjroism eðni thleskopikì. Prthr ste epðshc ìti (7..) φ(x ) = R n,f,x (x) ki φ(x) = R n,f,x (x) =. (i) Gi thn morf Cuchy tou upoloðpou efrmìzoume to Je rhm Mèshc Tim c gi thn φ sto diˆsthm me ˆkr x ki x : Upˆrqei ξ metxô twn x ki x ste Apì thn èpeti ìti R n,f,x (x) = φ(x ) φ(x) = φ (ξ)(x x). φ (ξ) = f (n+) (ξ) (x ξ) n n! R n,f,x (x) = f (n+) (ξ) (x ξ) n (x x ). n! (ii) Gi thn morf Lgrnge tou upoloðpou efrmìzoume to Je rhm Mèshc Tim c tou Cuchy gi thn φ ki gi thn g(t) = (x t) n+ sto diˆsthm me ˆkr x ki x : Upˆrqei ξ metxô twn x ki x ste 'Epeti ìti R n,f,x (x) (x x ) n+ = φ(x ) φ(x) g(x ) g(x) = φ (ξ) g (ξ). R n,f,x (x) = f (n+) (ξ) n! (x ξ) n (n + )(x ξ) n (x x ) n+ = f (n+) (ξ) (n + )! (x x ) n+. (iii) Gi thn oloklhrwtik morf tou upoloðpou prthroôme ìti (pì thn upìjes mc) h φ eðni oloklhr simh sto diˆsthm me ˆkr x ki x, opìte efrmìzeti to deôtero jemeli dec je rhm tou ApeirostikoÔ LogismoÔ: R n,f,x (x) = φ(x ) φ(x) = x = = x x x x x φ (t) dt f (n+) (t) (x t) n dt n! f (n+) (t) (x t) n dt. n! 'Etsi, èqoume tic treic morfèc gi to upìloipo R n,f,x (x). Sthn epìmenh Prˆgrfo j qrhsimopoi soume to Je rhm Tylor gi n broume to nˆptugm se dunmoseirˆ twn bsik n uperbtik n sunrt sewn.

6 Je rhm Tylor 7. Dunmoseirèc ki nptôgmt Tylor 7.þ H ekjetik sunˆrthsh f(x) = e x PrthroÔme ìti f (k) (x) = e x gi kˆje x R ki k =,,,.... f (k) () = gi kˆje k. Sunep c, (7..) T n (x) := T n,f, (x) = n x k k! = + x + x! + + xn n!. 'Estw x. Qrhsimopoi ntc thn morf Lgrnge tou upoloðpou pðrnoume (7..) R n (x) := R n,f, (x) = e ξ (n + )! xn+ Eidikìter, gi kˆpoio ξ metxô twn ki x. Gi n ektim soume to upìloipo dikrðnoume dôo peript seic. An x > tìte R n (x) = An x <, tìte ξ < ki e ξ <, ˆr Se kˆje perðptwsh, R n (x) = e ξ (n + )! xn+ ex x n+ (n + )!. e ξ (n + )! x n+ x n+ (n + )!. (7..3) R n (x) e x x n+ (n + )!. 'Estw x. Efrmìzontc to krit rio tou lìgou gi thn koloujð n := e x x n+ (n+)! blèpoume ìti n+ = x n n +, ˆr lim R n(x) =. n Sunep c, (7..4) e x = lim n T n(x) = gi kˆje x R. 7.bþ H sunˆrthsh f(x) = cos x PrthroÔme ìti f (k) (x) = ( ) k cos x ki f (k+) (x) = ( ) k sin x gi kˆje x R ki k =,,,.... Eidikìter, f (k) () = ( ) k ki f (k+) () =. Sunep c, (7..5) T n (x) := T n,f, (x) = n ( ) k x k (k)! x k k! = x! + x4 4! + ( )n x n. (n)!

7. Dunmoseirès ki nptôgmt Tylor 7 'Estw x. Qrhsimopoi ntc thn morf Lgrnge tou upoloðpou pðrnoume (7..6) R n (x) := R n,f, (x) = f (n+) (ξ) (n + )! xn+ gi kˆpoio ξ metxô twn ki x. Gi n ektim soume to upìloipo prthroôme ìti f (n+) (ξ) (eðni kˆpoio hmðtono sunhmðtono), ˆr (7..7) R n (x) x n+ (n + )!. Efrmìzontc to krit rio tou lìgou gi thn koloujð n := x n+ (n+)! blèpoume ìti Sunep c, lim R n(x) =. n (7..8) cos x = lim n T n(x) = gi kˆje x R. 7.gþ H sunˆrthsh f(x) = sin x ( ) k x k (k)! PrthroÔme ìti f (k) (x) = ( ) k sin x ki f (k+) (x) = ( ) k cos x gi kˆje x R ki k =,,,.... Eidikìter, f (k) () = ki f (k+) () = ( ) k. Sunep c, (7..9) T n+ (x) := T n+,f, (x) = n ( ) k x k+ (k + )! = x x3 3! + x5 5! + ( )n x n+ (n + )! 'Estw x. Qrhsimopoi ntc thn morf Lgrnge tou upoloðpou pðrnoume (7..) R n+ (x) := R n+,f, (x) = f (n+) (ξ) (n + )! xn+ gi kˆpoio ξ metxô twn ki x. Gi n ektim soume to upìloipo prthroôme ìti f (n+) (ξ) (eðni kˆpoio hmðtono sunhmðtono), ˆr (7..) R n+ (x) x n+ (n + )!. Efrmìzontc to krit rio tou lìgou gi thn koloujð n := x n+ (n+)! blèpoume ìti. Sunep c, lim R n+(x) =. n (7..) sin x = lim n T n+(x) = gi kˆje x R. ( ) k x k+ (k + )!

8 Je rhm Tylor 7.dþ H sunˆrthsh f(x) = ln( + x), x (, ] PrthroÔme ìti f (k) (x) = ( )k (k )! gi kˆje x > ki k =,,.... Eidikìter, f() = ki f (k) () = ( ) k (k )! gi kˆje k. (+x) k Sunep c, (7..3) T n (x) := T n,f, (x) = n ( ) k x k k = x x + x3 3 + ( )n x n n 'Estw x >. Qrhsimopoi ntc thn oloklhrwtik morf tou upoloðpou pðrnoume (7..4) R n (x) := R n,f, (x) = ( ) n x (x t) n dt. ( + t) n+ Jètoume u = x t. Tìte, to +t u metbˆlleti pì x wc ki dt Sunep c, (7..5) R n (x) = DikrÐnoume dôo peript seic: An < x < tìte R n (x) An < x tìte Se kˆje perðptwsh, Sunep c, x R n (x) = x u n + u du + x x u n + u du u n + u du. x x lim R n(x) =. n (7..6) ln( + x) = lim n T n(x) = +t = du +u u n du = x n+ + x n +. u n du = x n+ (n + ). ( ) k x k gi kˆje x (, ] (seirˆ Merctor). Eidikìter, gi x = pðrnoume ton tôpo tou Leibniz (7..7) ln = ( ) k DeÔteroc trìpoc: Apì th sqèsh (7..8) èqoume, gi kˆje x >, (7..9) ln( + x) = k = + 3 4 + + ( )n n + t = t + t + + ( ) n t n + ( ) n tn + t x k (elègxte to). +. (t ) x x dt = x + t + x3 xn + + ( )n 3 n + ( )n. t n + t dt.

7. Dunmoseirès ki nptôgmt Tylor 9 An onomˆsoume F n (x) th diforˆ ) (7..) ln( + x) (x x + x3 xn + + ( )n 3 n èqoume (7..) F n (x) = ( ) n x t n + t dt. Ektim ntc to olokl rwm ìpwc prin, blèpoume ìti { } x n+ (7..) F n (x) mx, x + n + gi kˆje < x. Sunep c, lim n F n (x) =. 'Epeti ìti (7..3) ln( + x) = ( ) n= n xn F gi x (, ]. Prthr ste epðshc ìti n (x) lim x x n =, to opoðo podeiknôei ìti F n (x) = R n,f, (x). 'Otn x > h seirˆ poklðnei (foô h koloujð ( ) x n n den teðnei sto ) ki gi x = epðshc poklðnei (rmonik seirˆ). 7.eþ H diwnumik sunˆrthsh f(x) = ( + x), x > H f orðzeti pì thn f(x) = exp( ln( + x)). An >, to ìrio lim x f(x) upˆrqei ki eðni Ðso me, diìti ln( + x) = y ki exp(y). Se ut n thn perðptwsh mporoôme n epekteðnoume to pedðo orismoô thc f sto [, ) jètontc f( ) =. PrgwgÐzontc blèpoume ìti Sunep c, f (k) (x) = ( ) ( k + )( + x) k f (k) () = ( ) ( k + ). (7..4) T n (x) := T n,f, (x) = n n ( ) x k k ìpou (7..5) ( ) = k ( ) ( k + ). k! Prthr ste ìti n N tìte ( k) = gi kˆje k N me k >, opìte (7..6) ( + x) = ( ) x k. k

Je rhm Tylor Upojètoume loipìn ìti / N. J deðxoume ìti, ìtn x <, tìte T n,f, (x) f(x). QrhsimopoioÔme th morf Cuchy tou upoloðpou: upˆrqei ξ nˆmes sto ki sto x ste R n (x) = f (n+) (ξ) (x ξ) n ( )... ( n) x = ( + ξ) (n+) (x ξ) n x n! ( ) n! n ( )... ( n) x ξ = ( + ξ) x n! + ξ Gi n deðxoume ìti lim R n(x) = ìtn x <, prthroôme pr t ìti n (7..7) x ξ + ξ x ìtn x <. Prˆgmti, n ξ x èqoume (7..8) x ξ + ξ = x ξ + ξ x + ξ x = x. An < x ξ jewroôme thn sunˆrthsh g x : [x, ] R me (7..9) g x (ξ) = x ξ + ξ = x + ξ + h opoð eðni fjðnous (foô x + > ) ˆr èqei mègisth tim thn g x (x) = ki elˆqisth thn g x () = x opìte gi kˆje t [x, ] èqoume g x (ξ) ˆr (7..3) x ξ + ξ = g x(ξ) g x () = x = x. 'Epeti ìti R n (x) = ( )... ( n) n! ( x ξ + ξ ) n ( + ξ) x ( )... ( n) x n ( + ξ) x n! ( )... ( n) x n M(x) n! ìpou M(x) = x mx(, ( + x) ) (ˆskhsh), ˆr rkeð n deðxoume ìti y n ( )... ( n) x n n! kj c n. 'Eqoume y n+ = ( )... ( n)( (n + )) y n ( )... ( n) n! (n + )! x n+ x n Gi kˆje n èqoume (n + ) = n +, ˆr y n+ (7..3) = (n + ) y n x n + = n + x x < n + = n + n + x.

7.3 Sunrt seis prstˆsimes se dunmoseirˆ ìtn n, ˆr y n. DeÐxme ìti n x < tìte lim n R n(x) =. 'Ar, (7..3) ( + x) = n= ( ) x k k gi < x <. Gi x > h seirˆ poklðnei (krit rio lìgou). Gi x = h sumperiforˆ exrtˆti pì thn tim tou. Gi prˆdeigm, ìtn =, h seirˆ poklðnei ki st dôo ˆkr (gewmetrik seirˆ me lìgo x). ApodeiknÔeti ìti ìtn = / h seirˆ sugklðnei gi x = ki poklðnei gi x =, ki ìtn = / (ki genikìter ìtn > ), h seirˆ sugklðnei ki st dôo ˆkr. 7. þ H sunˆrthsh f(x) = rctn x, x Xekinˆme pì thn rctn x = x + t dt. AntÐ n prgwgðsoume n forèc thn rctn sto, eðni eukolìtero n oloklhr soume thn (7..33) opìte (7..34) An orðsoume x + t = t + t 4 t 6 + + ( ) n t n + ( )n+ t n+ + t x x3 xn+ dt = x + + ( )n + t 3 n + + t n+ ( )n+ + t dt. (7..35) p n (x) = x x3 3 èqoume (7..36) f(x) p n (x) = + + ( )n xn+ n + x t n+ + t dt x opìte, ìtn x, blèpoume ìti lim n p n(x) = f(x), dhld (7..37) rctn x = n= ( ) n xn+ t n+ dt x n+3 n + 3 n + = x x3 3 + x5 5 + gi x [, ]. 7.3 Sunrt seic prstˆsimec se dunmoseirˆ Sthn Prˆgrfo.4 suzht sme gi pr th forˆ tic dunmoseirèc. EÐdme ìti n k x k eðni mi dunmoseirˆ me suntelestèc k, tìte to sônolo twn shmeðwn st

Je rhm Tylor opoð sugklðnei h dunmoseirˆ eðni {ousistikˆ} èn diˆsthm summetrikì wc proc to (, endeqomènwc, to {} to R). An orðsoume R := sup{ x : h dunmoseirˆ sugklðnei sto x}, tìte h dunmoseirˆ sugklðnei polôtwc se kˆje x ( R, R) ki poklðnei se kˆje x me x > R. To diˆsthm ( R, R) onomˆzeti diˆsthm sôgklishc thc dunmoseirˆc. To sônolo sôgklishc thc dunmoseirˆc, dhld to sônolo ìlwn twn shmeðwn st opoð sugklðnei, prokôptei pì to ( R, R) me thn prosj kh (Ðswc) tou R tou R twn ±R. Sthn perðptwsh pou R = +, h dunmoseirˆ sugklðnei se kˆje x R. Sthn perðptwsh pou R =, h dunmoseirˆ sugklðnei mìno sto shmeðo x =. H epìmenh Prìtsh dðnei ènn {tôpo} gi thn ktðn sôgklishc. Prìtsh 7.3.. 'Estw sôgklis c thc dðneti pì thn k x k mi dunmoseirˆ me suntelestèc k. H ktðn R = lim sup k /k. Apìdeixh. DeÐqnoume pr t ìti n x lim sup k k k <, tìte h seirˆ k x k sugklðnei polôtwc. Prˆgmti, n jewr soume s > me x lim sup k k < s <, tìte pì ton qrkthrismì tou lim sup, upˆrqei N N ste k x k s k gi kˆje k N, ki to sumpèrsm èpeti pì to krit rio sôgkrishc. k >, tìte h seirˆ k Sth sunèqei deðqnoume ìti n x lim sup k poklðnei. Prˆgmti, n jewr soume s > me x lim sup k k > s >, tìte pì ton k qrkthrismì tou lim sup, upˆrqoun ˆpeiroi deðktec k < k < < k n < k n+ < ste kn x kn s kn > gi kˆje n N. 'Ar, k x k ki efrmìzeti to krit rio pìklishc. Sunduˆzontc t prpˆnw blèpoume ìti to diˆsthm sôgklishc thc dunmoseirˆc eðni ngkstikˆ to ( R, R), ìpou R = lim sup k /k. Orismìc 7.3.. Lème ìti mi sunˆrthsh f : ( R, R) R eðni prstˆsimh se dunmoseirˆ me kèntro to n upˆrqei koloujð ( k ) prgmtik n rijm n ste f(x) = k x k gi kˆje x ( R, R). To je rhm pou koloujeð deðqnei ìti n mi sunˆrthsh eðni prstˆsimh se dunmoseirˆ sto ( R, R), tìte eðni ˆpeirec forèc prgwgðsimh ki oi prˆgwgoð thc upologðzonti me prg gish twn ìrwn thc dunmoseirˆc. Anˆlog, upologðzeti to olokl rwmˆ thc se kˆje upodiˆsthm tou ( R, R). Je rhm 7.3.3 (je rhm prg gishc dunmoseir n). 'Estw n= nx n mi dunmoseirˆ pou sugklðnei sto ( R, R) gi kˆpoion R >. JewroÔme th sunˆrthsh f : ( R, R) R me f(x) = k x k.

7.3 Sunrt seis prstˆsimes se dunmoseirˆ 3 Tìte, h f eðni ˆpeirec forèc prgwgðsimh: gi kˆje k ki gi kˆje x < R isqôei f (k) (x) = n(n ) (n k + ) n x n k. EpÐshc, n=k n = f (n) (), n =,,,... n! ki, gi kˆje x < R, h f eðni oloklhr simh sto [, x] ki x f(t) dt = n= n n + xn+. Apìdeixh. DeÐqnoume pr t ìti, gi kˆje x ( R, R), () f (x) = n n x n. AfoÔ x < R, upˆrqei δ > ste x + δ < R. 'Epeti (exhg ste gitð) ìti n= n ( x + δ) n < +. n= 'Estw < t < δ. Prthr ste ìti (x + t) n x n nx n t = Sunep c, f(x + t) f(x) t n ( n k k= n t δ k= ) x n k t k = t n ( n k t δ ( x + δ)n. n n x n = n= PÐrnontc to ìrio kj c to t, blèpoume ìti to opoðo podeiknôei thn (). f(x + t) f(x) lim = t t k= ) x n k t k δ t ( ) n x n k t k k n δ k= n= t δ n ( x + δ) n. ( ) n x n k δ k k n (x + t) n x n nx n t t n= n n x n, n=

4 Je rhm Tylor Qrhsimopoi ntc thn Prìtsh 7.3. blèpoume ìti h dunmoseirˆ n= n nx n èqei thn Ðdi ktðn sôgklishc me thn n= nx n (exhg ste gitð). Efrmìzontc loipìn ton Ðdio sullogismì gi thn f sth jèsh thc f, blèpoume ìti f () (x) = n(n ) n x n. n= SuneqÐzontc me ton Ðdio trìpo, podeiknôoume ìti h f eðni ˆpeirec forèc prgwgðsimh ki ìti gi kˆje k ki gi kˆje x < R isqôei () f (k) (x) = n(n ) (n k + ) n x n k. n=k Jètontc x = sthn () blèpoume ìti f (k) () = k! k gi kˆje k (prthr ste ìti: n jèsoume x = sto dexiì mèloc thc (), tìte ìloi oi ìroi tou jroðsmtoc mhdenðzonti, ektìc pì ekeðnon pou ntistoiqeð sthn tim n = k ki isoôti me k(k ) k x = k! k ). Gi ton teleutðo isqurismì, prthroôme ìti h dunmoseirˆ n n= n+ xn+ èqei thn Ðdi ktðn sôgklishc me thn n= nx n (exhg ste gitð) ki prgwgðzontc ìro proc ìro thn n F (x) = n + xn+ sto ( R, R) pðrnoume F (x) = n= n x n = f(x). n= Apì to jemeli dec je rhm tou ApeirostikoÔ LogismoÔ èpeti ìti x n f(t) dt = F (x) F () = F (x) = n + xn+ gi kˆje x ( R, R). Pìrism 7.3.4 (je rhm mondikìthtc). 'Estw ( k ), (b k ) koloujðec prgmtik n rijm n me thn ex c idiìtht: upˆrqei R > ste k x k = b k x k gi kˆje x ( R, R). Tìte, n= k = b k gi kˆje k =,,,.... Apìdeixh. Apì to Je rhm 7.3.3, gi th sunˆrthsh f : ( R, R) R me f(x) = k x k = b k x k èqoume f (k) () = k! k = k!b k gi kˆje k. Sunep c, k = b k gi kˆje k.

7.4 Ask seic Pr th Omˆd 7.4 Ask seis 5. 'Estw p(x) = + x + + n x n polu numo bjmoô n ki èstw R. DeÐxte ìti upˆrqoun b, b,, b n R ste p(x) = b + b (x ) + + b n (x ) n gi kˆje x R. DeÐxte ìti b k = p(k) (), k =,,..., n. k!. Grˆyte kjèn pì t prkˆtw polu num sth morf b + b (x 3) + + b n (x 3) n : p (x) = x 4x 9, p (x) = x 4 x 3 + 44x + x +, p 3 (x) = x 5. 3. Gi kˆje mð pì tic prkˆtw sunrt seic, n brejeð to polu numo Tylor T n,f, pou upodeiknôeti. (T 3,f, ) : f(x) = exp(sin x). (T n+,f, ) : f(x) = ( + x ). (T n,f, ) : f(x) = ( + x). (T 4,f, ) : f(x) = x 5 + x 3 + x. (T 6,f, ) : f(x) = x 5 + x 3 + x. (T 5,f, ) : f(x) = x 5 + x 3 + x. 4. 'Estw n ki f, g : (, b) R sunrt seic n forèc prgwgðsimec sto x (, b) ste f(x ) = f (x ) = = f (n ) (x ) =, g(x ) = g (x ) = = g (n ) (x ) = ki g (n) (x ). DeÐxte ìti f(x) lim x x g(x) = f (n) (x ) g (n) (x ). 5. 'Estw n ki f : (, b) R sunˆrthsh n forèc prgwgðsimh sto x (, b) ste f(x ) = f (x ) = = f (n ) (x ) = ki f (n) (x ). DeÐxte ìti: () An o n eðni ˆrtioc ki f (n) (x ) >, tìte h f èqei topikì elˆqisto sto x. (b) An o n eðni ˆrtioc ki f (n) (x ) <, tìte h f èqei topikì mègisto sto x. (g) An o n eðni perittìc, tìte h f den èqei topikì mègisto oôte topikì elˆqisto sto x, llˆ to x eðni shmeðo kmp c gi thn f. 6. An f(x) = ln x, x >, breðte thn plhsièsterh eujeð ki thn plhsièsterh prbol sto grˆfhm thc f sto shmeðo (e, ). 7. BreÐte to polu numo Tylor T n,f, gi th sunˆrthsh f(x) = x e t dt, (x R).

6 Je rhm Tylor 8. BreÐte to polu numo Tylor T n,f, gi th sunˆrthsh f : R R pou orðzeti wc ex c: f() = ki f(x) = e /x, x. 9. Qrhsimopoi ntc to nˆptugm Tylor thc sunˆrthshc rctn x ( x ) upologðste to ˆjroism ( ) n 3 n (n + ). n=. 'Estw f : R R ˆpeirec forèc prgwgðsimh sunˆrthsh. Upojètoume ìti f = f ki f() =, f () = f () =. () 'Estw R >. DeÐxte ìti upˆrqei M = M(R) > ste: gi kˆje x [ R, R] ki gi kˆje k =,,,..., f (k) (x) M. (b) BreÐte to polu numo Tylor T 3n,f, ki, qrhsimopoi ntc to () ki opoiond pote tôpo upoloðpou, deðxte ìti x 3k f(x) = (3k)! gi kˆje x R.. BreÐte proseggistik tim, me sfˆlm mikrìtero tou 6, gi kjènn pì touc rijmoôc sin, sin, sin, e, e.. () DeÐxte ìti π 4 = rctn + rctn 3 ki π 4 = 4 rctn 5 rctn 39. (b) DeÐxte ìti π = 3.459 (me ˆll lìgi, breðte proseggistik tim gi ton rijmì π me sfˆlm mikrìtero tou 6 ).

Kefˆlio 8 Kurtèc ki koðlec sunrt seic 8. Orismìc Se utì to kefˆlio, me I sumbolðzoume èn (kleistì, noiktì hminoiktì, pepersmèno ˆpeiro) diˆsthm sto R. 'Estw, b R me < b. Sto epìmeno L mm perigrˆfoume t shmeð tou eujôgrmmou tm mtoc [, b]. L mm 8... An < b sto R tìte (8..) [, b] = {( t) + tb : t }. Eidikìter, gi kˆje x [, b] èqoume (8..) x = b x b + x b b. Apìdeixh. EÔkol elègqoume ìti, gi kˆje t [, ] isqôei (8..3) ( t) + tb = + t(b ) b, dhld (8..4) {( t) + tb : t } [, b]. AntÐstrof, kˆje x [, b] grˆfeti sth morf (8..5) x = b x b + x b b. Prthr ntc ìti t := (x )/(b ) [, ] ki t = (b x)/(b ), blèpoume ìti (8..6) [, b] {( t) + tb : t }. T shmeð ( t) + tb tou [, b] lègonti kurtoð sundusmoð twn ki b.

8 Kurtès ki koðles sunrt seis Orismìc 8... 'Estw f : I R mi sunˆrthsh. () H f lègeti kurt n (8..7) f(( t) + tb) ( t)f() + tf(b) gi kˆje, b I ki gi kˆje t R me < t < (prthr ste ìti, foô to I eðni diˆsthm, to L mm 8.. deðqnei ìti to shmeðo ( t) + tb [, b] I, dhld h f orðzeti klˆ se utì). H gewmetrik shmsð tou orismoô eðni h ex c: h qord pou èqei sn ˆkr t shmeð (, f()) ki (b, f(b)) den eðni poujenˆ kˆtw pì to grˆfhm thc f. (b) H f lègeti gnhsðwc kurt n eðni kurt ki èqoume gn si nisìtht sthn (8..7) gi kˆje < b sto I ki gi kˆje < t <. (g) H f : I R lègeti koðlh (ntðstoiq, gnhsðwc koðlh) n h f eðni kurt (ntðstoiq, gnhsðwc kurt ). Prt rhsh 8..3. IsodÔnmoi trìpoi me touc opoðouc mporeð n perigrfeð h kurtìtht thc f : I R eðni oi ex c: () An, b, x I ki < x < b, tìte (8..8) f(x) b x b f() + x b f(b). Prthr ste ìti to dexiì mèloc ut c thc nisìthtc isoôti me (8..9) f() + f(b) f() (x ). b (b) An, b I ki n t, s > me t + s =, tìte (8..) f(t + sb) tf() + sf(b). 8. Kurtèc sunrt seic orismènec se noiktì diˆsthm Se ut thn Prˆgrfo meletˆme wc proc th sunèqei ki thn prgwgisimìtht mi kurt sunˆrthsh pou orðzeti se noiktì diˆsthm. 'Ol t potelèsmt pou j podeðxoume eðni sunèpeiec tou kìloujou {l mmtoc twn tri n qord n}: Prìtsh 8.. (to l mm twn tri n qord n). 'Estw f : (, b) R kurt sunˆrthsh. An y < x < z sto (, b), tìte (8..) f(x) f(y) x y Apìdeixh. AfoÔ h f eðni kurt, èqoume f(z) f(y) z y f(z) f(x). z x (8..) f(x) z x z y f(y) + x y z y f(z). Apì ut thn nisìtht blèpoume ìti (8..3) f(x) f(y) y x z y f(y) + x y z y f(z) = x y [f(z) f(y)], z y

8. Kurtès sunrt seis orismènes se noiktì diˆsthm 9 to opoðo podeiknôei thn rister nisìtht sthn (8..). Xekin ntc pˆli pì thn (8..), grˆfoume (8..4) f(x) f(z) z x z y f(y) + x z x f(z) = z [f(z) f(y)], z y z y p ìpou prokôptei h dexiˆ nisìtht sthn (8..). J qrhsimopoi soume epðshc thn ex c pl sunèpei tou l mmtoc twn tri n qord n. L mm 8... 'Estw f : (, b) R kurt sunˆrthsh. An y < x < z < w sto (, b), tìte (8..5) f(x) f(y) x y f(w) f(z). w z Apìdeixh. Efrmìzontc thn Prìtsh 8.. gi t shmeð y < x < z, pðrnoume (8..6) f(x) f(y) x y f(z) f(x). z x Efrmìzontc pˆli thn Prìtsh 8.. gi t shmeð x < z < w, pðrnoume (8..7) 'Epeti to sumpèrsm. f(z) f(x) z x f(w) f(z). w z Je rhm 8..3. 'Estw f : (, b) R kurt sunˆrthsh. upˆrqoun oi pleurikèc prˆgwgoi (8..8) f (x) = lim h f(x + h) f(x) h ki An x (, b), tìte f +(x) f(x + h) f(x) = lim. h + h Apìdeixh. J deðxoume ìti upˆrqei h dexiˆ pleurik prˆgwgoc f +(x) (me ton Ðdio trìpo douleôoume gi thn rister pleurik prˆgwgo f (x)). JewroÔme th sunˆrthsh g x : (x, b) R pou orðzeti pì thn (8..9) g x (z) := f(z) f(x). z x H g x eðni Ôxous: n x < z < z < b, to l mm twn tri n qord n deðqnei ìti (8..) g x (z ) = f(z ) f(x) z x f(z ) f(x) z x = g x (z ). EpÐshc, n jewr soume tuqìn y (, x), to l mm twn tri n qord n (gi t y < x < z) deðqnei ìti (8..) f(x) f(y) x y f(z) f(x) z x = g x (z) gi kˆje z (x, b), dhld h g x eðni kˆtw frgmènh. 'Ar, upˆrqei to (8..) lim g f(z) f(x) f(x + h) f(x) x(z) = lim = lim. z x + z x + z x h + h Dhld, upˆrqei h dexiˆ pleurik prˆgwgoc f +(x).

3 Kurtès ki koðles sunrt seis Je rhm 8..4. 'Estw f : (, b) R kurt sunˆrthsh. Oi pleurikèc prˆgwgoi f, f + eðni Ôxousec sto (, b) ki f f + sto (, b). Apìdeixh. 'Estw x < y sto (, b). Gi rketˆ mikrì jetikì h èqoume x ± h, y ± h (, b) ki x + h < y h. Apì thn Prìtsh 8.. ki pì to L mm 8.. blèpoume ìti (8..3) f(x) f(x h) f(x + h) f(x) f(y) f(y h) h h h PÐrnontc ìri kj c h +, sumperðnoume ìti (8..4) f (x) f +(x) f (y) f +(y). f(y + h) f(y). h Oi nisìthtec f (x) f (y) ki f +(x) f +(y) deðqnoun ìti oi f, f + eðni Ôxousec sto (, b). H rister nisìtht sthn (8..4) deðqnei ìti f f + sto (, b). H Ôprxh twn pleurik n prg gwn exsflðzei ìti kˆje kurt sunˆrthsh f : I R eðni suneq c sto eswterikì tou I: Je rhm 8..5. Kˆje kurt sunˆrthsh f : (, b) R eðni suneq c. Apìdeixh. 'Estw x (, b). Tìte, gi mikrˆ h > èqoume x + h, x h (, b) ki (8..5) f(x + h) = f(x) + ìtn h +, en, teleðwc nˆlog, (8..6) f(x h) = f(x) + f(x + h) f(x) h f(x h) f(x) h ìtn h +. 'Ar, h f eðni suneq c sto x. 8.3 PrgwgÐsimec kurtèc sunrt seic h f(x) + f +(x) = f(x) ( h) f(x) + f (x) = f(x) Ston Apeirostikì Logismì I dìjhke ènc diforetikìc orismìc thc kurtìthtc gi mi prgwgðsimh sunˆrthsh f : (, b) R. Gi kˆje x (, b), jewr sme thn efptomènh (8.3.) u = f(x) + f (x)(u x) tou grf mtoc thc f sto (x, f(x)) ki eðpme ìti h f eðni kurt sto (, b) n gi kˆje x (, b) ki gi kˆje y (, b) èqoume (8.3.) f(y) f(x) + f (x)(y x). Dhld, n to grˆfhm {(y, f(y)) : < y < b} brðsketi pˆnw pì kˆje efptomènh. To epìmeno je rhm deðqnei ìti, n perioristoôme sthn klˆsh twn prgwgðsimwn sunrt sewn, oi {dôo orismoð} sumfwnoôn.

8.3 PrgwgÐsimes kurtès sunrt seis 3 Je rhm 8.3.. 'Estw f : (, b) R prgwgðsimh sunˆrthsh. T ex c eðni isodônm: () H f eðni kurt. (b) H f eðni Ôxous. (g) Gi kˆje x, y (, b) isqôei h (8.3.3) f(y) f(x) + f (x)(y x). Apìdeixh. Upojètoume ìti h f eðni kurt. AfoÔ h f eðni prgwgðsimh, èqoume f = f = f + sto (, b). Apì to Je rhm 8..4 oi f, f + eðni Ôxousec, ˆr h f eðni Ôxous. Upojètoume t r ìti h f eðni Ôxous. 'Estw x, y (, b). An x < y, efrmìzontc to je rhm mèshc tim c sto [x, y], brðskoume ξ (x, y) ste f(y) = f(x) + f (ξ)(y x). AfoÔ ξ > x ki h f eðni Ôxous, èqoume f (ξ) f (x). AfoÔ y x >, èpeti ìti (8.3.4) f(y) = f(x) + f (ξ)(y x) f(x) + f (x)(y x). An x > y, efrmìzontc to je rhm mèshc tim c sto [y, x], brðskoume ξ (y, x) ste f(y) = f(x) + f (ξ)(y x). AfoÔ ξ < x ki h f eðni Ôxous, èqoume f (ξ) f (x). AfoÔ y x <, èpeti pˆli ìti (8.3.5) f(y) = f(x) + f (ξ)(y x) f(x) + f (x)(y x). Tèloc, upojètoume ìti h (8.3.3) isqôei gi kˆje x, y (, b) ki j deðxoume ìti h f eðni kurt. 'Estw x < y sto (, b) ki èstw < t <. Jètoume z = ( t)x + ty. Efrmìzontc thn upìjesh gi t zeugˆri x, z ki y, z, pðrnoume (8.3.6) f(x) f(z) + f (z)(x z) ki f(y) f(z) + f (z)(y z). 'Ar, ( t)f(x) + tf(y) ( t)f(z) + tf(z) + f (z)[( t)(x z) + t(y z)] = f(z) + f (z)[( t)x + ty z] = f(z). Dhld, ( t)f(x) + tf(y) f(( t)x + ty). Sthn perðptwsh pou h f eðni dôo forèc prgwgðsimh sto (, b), h isodunmð twn () ki (b) sto Je rhm 8.3. dðnei ènn plì qrkthrismì thc kurtìthtc mèsw thc deôterhc prg gou. Je rhm 8.3.. 'Estw f : (, b) R dôo forèc prgwgðsimh sunˆrthsh. H f eðni kurt n ki mìno n f (x) gi kˆje x (, b). Apìdeixh. H f eðni Ôxous n ki mìno n f sto (, b). 'Omwc, sto Je rhm 8.3. eðdme ìti h f eðni Ôxous n ki mìno n h f eðni kurt.

3 Kurtès ki koðles sunrt seis 8.4 Anisìtht tou Jensen H nisìtht tou Jensen podeiknôeti me epgwg ki {genikeôei} thn nisìtht tou orismoô thc kurt c sunˆrthshc. Prìtsh 8.4. (nisìtht tou Jensen). 'Estw f : I R kurt sunˆrthsh. An x,..., x m I ki t,..., t m me t + + t m =, tìte m i= t ix i I ki (8.4.) f(t x + + t m x m ) t f(x ) + + t m f(x m ). Apìdeixh. 'Estw = min{x,..., x m } ki b = mx{x,..., x m }. AfoÔ to I eðni diˆsthm ki, b I, sumperðnoume ìti {x,..., x m } [, b] I. AfoÔ t i ki t + + t m =, èqoume (8.4.) = (t + + t m ) t x + + t m x m (t + + t m )b = b, dhld, t x + + t m x m I. J deðxoume thn (8.4.) me epgwg wc proc m. Gi m = den èqoume tðpot n deðxoume, en gi m = h (8.4.) iknopoieðti pì ton orismì thc kurt c sunˆrthshc. Gi to epgwgikì b m upojètoume ìti m, x,..., x m, x m+ I ki t,..., t m, t m+ me t + +t m +t m+ =. MporoÔme n upojèsoume ìti kˆpoioc t i < (lli c, h nisìtht isqôei tetrimmèn). QwrÐc periorismì thc genikìthtc upojètoume ìti t m+ <. Jètoume t = t + + t m = t m+ >. AfoÔ x,..., x m I ki t t + + tm t =, h epgwgik upìjesh mc dðnei (8.4.3) x = t t x + + t m t x m I ki (8.4.4) tf(x) = tf ( t t x + + t ) m t x m t f(x ) + + t m f(x m ). Efrmìzontc t r ton orismì thc kurt c sunˆrthshc, pðrnoume (8.4.5) f(tx + t m+ x m+ ) tf(x) + t m+ f(x m+ ). Sunduˆzontc tic dôo prohgoômenec nisìthtec, èqoume f(t x + + t m x m + t m+ x m+ ) = f(tx + t m+ x m+ ) t f(x ) + + t m f(x m ) +t m+ f(x m+ ). Qrhsimopoi ntc thn nisìtht tou Jensen j deðxoume kˆpoiec klsikèc nisìthtec. H pr th pì utèc genikeôei thn nisìtht rijmhtikoô-gewmetrikoô mèsou. Anisìtht rijmhtikoô-gewmetrikoô mèsou. 'Estw x,..., x n ki r,..., r n jetikoð prgmtikoð rijmoð me r + + r n =. Tìte, (8.4.6) n i= x ri i n r i x i. i=

8.4 Anisìtht tou Jensen 33 Apìdeixh. H sunˆrthsh x ln x eðni koðlh sto (, + ). AfoÔ r i > ki r + + r n =, h nisìtht tou Jensen (gi thn kurt sunˆrthsh ln) deðqnei ìti (8.4.7) r ln x + + r n ln x n ln(r x + + r n x n ). Dhld, (8.4.8) ln(x r xrn n ) ln(r x + + r n x n ). To zhtoômeno prokôptei ˆmes pì to gegonìc ìti h ekjetik sunˆrthsh x e x eðni Ôxous. Eidikèc peript seic thc prohgoômenhc nisìthtc eðni oi ex c: () H klsik nisìtht rijmhtikoô-gewmetrikoô mèsou (8.4.9) (x x n ) /n x + + x n n ìpou x,..., x n >, h opoð prokôptei pì thn (8.4.6) n pˆroume r = = r n = n. (b) H nisìtht tou Young: An x, y > ki t, s > me t + s =, tìte (8.4.) x t y s tx + sy. H (8.4.) emfnðzeti polô suqnˆ sthn ex c morf : n x, y > ki p, q > me =, tìte p + q (8.4.) xy xp p + yq q Prˆgmti, rkeð n pˆroume touc x p, y q sth jèsh twn x, y ki touc p, q sth jèsh twn t, s. Oi p ki q lègonti suzugeðc ekjètec. Qrhsimopoi ntc ut thn nisìtht mporoôme n deðxoume thn klsik nisìtht tou Hölder: 'Estw p, q suzugeðc ekjètec. An,..., n ki b,..., b n eðni jetikoð prgmtikoð rijmoð, tìte (8.4.) ( n n i b i ) /p ( n ) /q b q i. p i i= i= i= Apìdeixh. Jètoume A = ( n i= p i )/p, B = ( n i= bq i )/q ki x i = i /A, y i = b i /B. Tìte, h zhtoômenh nisìtht (8.4.) pðrnei th morf (8.4.3) Apì thn (8.4.) èqoume (8.4.4) PrthroÔme ìti (8.4.5) n x i y i i= n i= n x i y i. i= ( x p ) i p + yq i = q p n x p i + q i= n y q i. i= n x p i = n n A p p i = ki y q i = n B q b q i =. i= i= i= i=

34 Kurtès ki koðles sunrt seis 'Ar, (8.4.6) n x i y i p + q =. i= 'Epeti h (8.4.).. Epilègontc p = q = pðrnoume thn nisìtht Cuchy-Schwrz: n,..., n ki b,..., b n eðni jetikoð prgmtikoð rijmoð, tìte (8.4.7) 8.5 Ask seic Omˆd A' ( n n i b i i= i= i ) / ( n ) / b i.. 'Estw f, f n : I R. Upojètoume ìti kˆje f n eðni kurt sunˆrthsh ki ìti f n (x) f(x) gi kˆje x I. DeÐxte ìti h f eðni kurt.. 'Estw {f n : n N} koloujð kurt n sunrt sewn f n : I R. OrÐzoume f : I R me f(x) = sup{f n (x) : n N}. An h f eðni pepersmènh pntoô sto I, tìte h f eðni kurt. 3. 'Estw f, g : R R kurtèc sunrt seic. Upojètoume kìm ìti h g eðni Ôxous. DeÐxte ìti h g f eðni kurt. 4. 'Estw f : I R kurt sunˆrthsh. DeÐxte ìti i= f(x + δ) f(x ) f(x + δ) f(x ) gi kˆje x < x I ki δ > gi to opoðo x + δ, x + δ I. 5. 'Estw f : [, b] R kurt sunˆrthsh. DeÐxte me èn prˆdeigm ìti h f den eðni ngkstikˆ sunˆrthsh Lipschitz se olìklhro to [, b], kìm ki n upojèsoume ìti h f eðni frgmènh. EpÐshc, deðxte ìti h f den eðni ngkstikˆ suneq c sto [, b]. 6. 'Estw f : (, b) R kurt sunˆrthsh ki ξ (, b). DeÐxte ìti: () n h f èqei olikì mègisto sto ξ tìte h f eðni stjer. (b) n h f èqei olikì elˆqisto sto ξ tìte h f eðni fjðnous sto (, ξ) ki Ôxous sto (ξ, b). (g) n h f èqei topikì elˆqisto sto ξ tìte èqei olikì elˆqisto sto ξ. (d) n h f eðni gnhsðwc kurt, tìte èqei to polô èn shmeðo olikoô elqðstou. 7. 'Estw f : R R kurt sunˆrthsh. An h f eðni ˆnw frgmènh, tìte eðni stjer. 8. DeÐxte ìti kˆje kurt sunˆrthsh orismènh se frgmèno diˆsthm eðni kˆtw frgmènh.

8.5 Ask seis 35 9. 'Estw f : (, + ) R koðlh, Ôxous, ˆnw frgmènh ki prgwgðsimh sunˆrthsh. DeÐxte ìti lim x + xf (x) =. Omˆd B'. DeÐxte ìti n h f : (, + ) R eðni kurt ki x,..., x m, y,..., y m >, tìte ( ) y + + y m m ( ) yi (x + + x m )f x i f. x + + x m x i DeÐxte ìti h f(x) = (+x p ) /p eðni kurt sto (, + ) ìtn p, ki sumperˆnte ìti m ((x + + x m ) p + (y + + y m ) p ) /p (x p i + yp i )/p.. DeÐxte ìti h sunˆrthsh sin x eðni kurt sto [, π]. Qrhsimopoi ntc to deðxte ìti h mègisth perðmetroc n-g nou pou eggrˆfeti sto mondiðo kôklo eðni n sin(π/n).. 'Estw α, α,..., α n jetikoð rijmoð. DeÐxte ìti ( ( + α )( + α ) ( + α n ) + (α α α n ) /n) n. i= i= [Upìdeixh: Prthr ste ìti h x ln( + e x ) eðni kurt.] Omˆd G' 3. 'Estw f : I R jetik koðlh sunˆrthsh. DeÐxte ìti h /f eðni kurt. 4. 'Estw f : [, π] R kurt sunˆrthsh. DeÐxte ìti gi kˆje k, π π f(x) cos kxdx. 5. 'Estw f : (, b) R suneq c sunˆrthsh. DeÐxte ìti h f eðni kurt n ki mìno n f(x) h f(x + t)dt h h gi kˆje diˆsthm [x h, x + h] (, b). 6. 'Estw f : (, b) R kurt sunˆrthsh ki c (, b). DeÐxte ìti h f eðni prgwgðsimh sto c n ki mìno n f(c + h) + f(c h) f(c) lim =. h + h 7. 'Estw f : [, + ) kurt, mh rnhtik sunˆrthsh me f() =. OrÐzoume F : [, + ) R me F () = ki DeÐxte ìti h F eðni kurt. F (x) = x x f(t)dt.