Katedra Informatiky Fakulta Matematiky, Fyziky a Informatiky Univerzita Komenského, Bratislava. Infinite series: Convergence tests (Bachelor thesis)

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1 Katedra Iformatiky Fakulta Matematiky, Fyziky a Iformatiky Uiverzita Komeského, Bratislava Ifiite series: Covergece tests Bachelor thesis) Frtišek Ďuriš Study programme: Iformatics Supervisor: doc. RNDr. Zbyěk Kubáček, CSc. Bratislava, 2009

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3 I hereby proclaim, that I worked out this bachelor thesis aloe d usig oly quoted sources

4 Author would like to thk his supervisor doc. RNDr. Zbyěk Kubáček, CSc. for his kid help d guidce, Bc. Peter Lečéš for extesive TEXical help, Bc. Peter Beňák for my advices d Jozef Vojtek for lguage support. 4

5 Thou, ature, art my goddess; to thy laws my services are boud... W. Shakespear, Kig Lear motto of J. C. F. Gauss 5

6 Abstract The theory of ifiite series, old d well examied part of calculus, gives us a powerfull tool for solvig a wide rge of problems such as evaluatig importt mathematical costts e, π,...), values of trigoometric fuctios etc. This thesis itroduces the idea of ifiite series with some basic theorems ecessary i the followig chapters. The buildig o this, it derives some covergece tests, amely Raabe s test, Gauss test, Bertrd s test d Kummer s test. It also proves that there is o uiversal compariso test for all series. Keywords: Bertrd, Raabe. Ifiite series, covergece, divergece, Kummer, Gauss, Abstrakt Teóriekoečých radov, stará a dobre preskúmá oblasť matematickej alýzy, ám dáva silý ástroj a riešeie širokého spektra problémov, ako apríklad vyčísleie zámych matematických koštát e, π,...), hodôt trigoometrických fukcii atď. Táto práca uvádza základé defiície a teorémy z tejto oblasti, ktoré budú dôležité v ďalších kapitolách. Potom, stavajúc a týchto základoch, odvádza vybré kritéria vyšetrujúce kovergeciu ekoečých radov, meovite Raabeho test, Gaussov test, Bertrdov test a Kummerov test. Tiež dokážeme, že eexistuje uiverzále porovávacie kritérium, ktoré by vedelo rohodúť kovergeciu/divergeciu všetkých radov. Kľúčové slová: Gauss, Bertrd, Raabe. Nekoečé rady, kovergecia, divergecia, Kummer, 6

7 Cotets Abstract 6 Cotets 7 Prologue 8. The historical backgroud of ifiite series Elemetary defiitios d theorems Joseph Ludwig Raabe 2 2. Raabe s ratio test Joh Carl Friedrich Gauss 7 3. Gauss test Joseph Louis Frcois Bertrd Bertrd s test O the uiversal criterio 30 6 Erst Eduard Kummer Kummer s test The Last chapter 4 8 Epilogue 50 Refereces 52 Bibliography 55 7

8 Chapter Prologue. The historical backgroud of ifiite series The first metio of ifiite series dates back to tiquity. I those times, to uderstd that a sum with ifiite umber of summds could have a fiite result was importt philosophical challege. Later, the theory of ifiite series was thoroughly developed d used to work out my sigifict problems that eluded solutios with y other approach. Greek Archimedes c. 287 BC c. 22 BC) was first kow mathematici who applied ifiite series to calculate the area uder the arc of parabola usig the method of exhaustio. Several ceturies later, Madhava c.350 c.425) from Idia came up with the idea of expdig fuctios ito ifiite series. He laid dow the precursors of moder coceptio of power series, Taylor series, Maclauri series, ratioal approximatios of ifiite series d ifiite cotiued fractios. Next importt step was take by Scotish mathematici James Gregory ). Gregory uderstood the differetial d itegral, before it was formulated by Newto d Leibiz, o such a level that he was able to fid the ifiite series of arctget by usig his ow methods. Eve though this is the mai result attributed to him, he discovered ifiite series The method of exhaustio is a method of fidig the area of a shape by iscribig iside it a sequece of polygos whose areas coverge to the area of the cotaiig shape. If the sequece is correctly costructed, the differece i area betwee the th polygo d the cotaiig shape will become arbitrarily small as becomes large. 2 2 adopted from of exhaustio o 2d May,

9 for tget, sect, arcsect d some others as well. After Eglish Sir Isaac Newto ) d Germ Gottfried Wilhelm Leibiz ) idepedetly developed d published formal methods of calculus, they achieved my discoveries withi the theory of ifiite series. But eve such a geius as Leibiz was uable to fid the sum of iverse squares This problem was evetually resolved by Swiss Leohard Euler ). Euler solved this problem usig ifiite series d developed ew ways to mipulate them. With the discovery of ifiite series there also came a questio which of them actually yield a fiite result. Obviously, the result of series k= k cot be fiite so how we c decide the more obscure oes? First mathematici to study covergece was Madhava i 4th cetury. He developed a test 3, which was further developed by his followers i the Kerala school. I Europe the developmet of covergece tests was started by Germ Joh Carl Friedrich Gauss ), but the terms of covergece d divergece had bee itroduced log before by J. Gregory. Frech Augusti Louis Cauchy ) proved that a product of two coverget series does ot have to be coverget, d with him begis the discovery of effective criterios, although his methods led to a special applicable for a certai rge of series) rather th a geeral criterios. Ad the same applies for Swiss Joseph Ludwig Raabe ), British Augustus De Morg ), Frech Joseph Louis Frcois Bertrd ) d others. Developmet of geeral criterios beg with Germ Erst Eduard Kummer ) d later was carried o by Germ Karl Theodor Wilhelm Weierstrass85-897), Germ Ferdid Eisestei ) d my others. 5 3 Early form of itegral test of covergece; i Europe it was later developed by Maclauri d Cauchy d is sometimes kow as the Maclauri - Cauchy test. 4 4 adopted from test for covergece o 2d May, text sources o 2d May, 2009): series 9

10 .2 Elemetary defiitios d theorems I this sectio we itroduce the basic defiitios such as what exactly the ifiite series are, what the covergece d divergece terms me etc. Also we will get acquaited with a few importt theorems which we will refer to i upcomig chapters. All defiitios were adopted from [].) Defiitio.2.. Let { } =k k N {0}) be a sequece. The we call the symbol =k or a k + a k+ + a k ) a ifiite) series. Moreover, as we cosider the symbols =k d =m +k m idetical, because we c write all series =k i the form = +k, all defiitios d theorems will be formulated for the series i the form = b. Defiitio.2.2. Let { } = be a sequece. The umber a k, resp. S k = a + a a k, k N is called k-th term, resp. k-th partial sum of series =. Defiitio.2.3. We say that the series = coverges is coverget) if there exists a fiite lim S. We call the umber lim S the sum of series = d we deote it with the same symbol = or a + a ). If the lim S is ot fiite the we say that the series = diverges is diverget) d we c distiguish three cases: ) if lim S = + we say that the series = diverges to + ; 2) if lim S = we say that the series = diverges to ; 3) if lim S does ot exist we say that the series = oscillate. Defiitio.2.4. =k+ is called k-th remaider of series =. 0

11 Havig become familiar with the most importt defiitios we c proceed to the formulatio of some theorems. All theorems were adopted from [2].) Theorem.2.5. The series = coverges if d oly if the followig is true: ɛ > 0, N N, N > N, p N p < ɛ. Remark.2.6. This theorem is kow as Cauchy-Bolzo s theorem. Corollary.2.7. Necessary requiremet for covergece) If the series = coverges the lim = 0. Corollary.2.8. If the series = coverges the ɛ > 0, k N : < ɛ. =k+ Theorem.2.9. Let at most fiite cout of umbers fail + b + b, where > 0, b > 0, = 0,, 2,... The the covergece of series = b implies the covergece of series = d the divergece of series = implies the divergece of series = b. Remark.2.0. Kow as the secod compariso test. Theorem.2.. Let y = fx) be a cotiuous, o-egative, o-icreasig fuctio defied o the iterval [, ). Let = f) for =, 2,... d F x) be a primitive fuctio to the fuctio fx) o the iterval [, a], where a > is arbitrary real umber. The: ) if lim x F x) is fiite the the series = coverges. 2) if lim x F x) = + the the series = diverges. Remark.2.2. Kow as the itegral test of covergece.

12 Chapter 2 Joseph Ludwig Raabe Bor o May 5th, 80, i Brody, Galicia, J. L. Raabe was a Swiss mathematici. He beg to study mathematics i 820 at the Polytechicum i Viea, Austria. I autum 83, he moved to Zurich, where he became a professor of mathematics i 833. I 855, he became a professor at the ewly fouded Swiss Polytechicum. Joseph Ludwig Raabe died o Juary 22d, 859, i Zurich, Switzerld. His best kow success is Raabe s test of covergece. biography source from 2d May, 2009): Ludwig Raabe 2

13 2. Raabe s ratio test Let s cosider the followig sequece = p p > 0 =, 2,... The first questio we may ask is wether the series = sometimes called Riem s series) coverges. At first glce we see that the ecessary requiremet for covergece lim = 0 holds, so we eed somethig more sophisticated. Let s put this sequece ito the itegral criterio d see what happes. First, we defie fx) as a fuctio fx) = x p o the iterval [, ), which satisfies f) = =, 2, 3,...). We check that the fuctio fx) meets all the precoditios required for this test d we cotiue by fidig a primitive fuctio F x) to the fuctio fx). After some easy calculatios, we have the result: { l x if p = F x) = if p p)x p ) Now we check the lim x F x) { lim F x) = K < if p > x if p 0, ] d accordig to the itegral test theorem.2.) we c coclude that the series = { p coverges if p > diverges if p 0, ] 2.) a This was easy, but let s go o d see what we c lear about +. This might tell us somethig about how fast approaches zero as teds to ifiity. Equatio follows: + = + p +) p = ) p = + ) p = + p ) + O 2 We used here + ) p = + p ) x x + O x )

14 as asymptotic approximatio valid for x. After some adjustmets we get ) = p + O + ) where the expresio O ) becomes isigifict compared to p as teds to ifiity. Thus for sufficietly large we c write ) p + Put together with 2.), we c formulate the followig theorem, which is kow as Raabe s test: Theorem 2... Let = be a series with positive terms k N : a k > 0). If ) p >, N N, > N : p 2.3) + the the series = coverges. If N N, > N : the the series = diverges. Proof. Because we c write p = q + ɛ, q >, ɛ > 0 d with 2.3) for sufficietly large ): ) q + ɛ q + O + + q ) + + O 2 = ) 2.4) + ) + ) q = q +) q Ad sice q >, the series = q coverges. Accordig to the secod compariso test theorem.2.9) the series = coverges as well. O the other hd, adjustig 2.4) we get + + = + Theorem.2.9 implies the divergece of series =. 4

15 A more geeral versio of this test is usig the remaider O 2 ), which we formerly hid iside the costt p): Theorem Let = be a series with positive terms. Assume that there exists a real bouded sequece B such that for all = + p + + B 2 The the series = coverges if d oly if p >. The series diverges if d oly if p. Proof. Let s assume that the series = coverges d p =. Usig Bertrd s test theorem 4.., see chapter four) l ) + With l Hopital s rule we get l ) = B l l 0 as teds to ifiity) d therefore the series = diverges accordig to Bertrd s test). A cotradictio. If p < the for sufficietly large ) ) + = p + B < d accordig to the theorem 2.. the series = diverges. A cotradictio. Thus p >. Now we assume that p >. For sufficietly large + = + p + B 2 + q, q > We fiish the proof with theorem 2... Divergece follows by alogy. Theorem Let = be a series with positive terms. Assume that there exists a real bouded sequece B such that for all = + p + + B 2 The the series = diverges if d oly if p. 5

16 Proof. Let s cosider a diverget series = d let p >. The for sufficietly large we get ) + = p + B = q, q > which implies accordig to the theorem 2..), that the series = is actually coverget. A cotradictio hece p ). Now for the part where p. Let us assume that p =, for the divergece i case p < is obvious. By usig Bertrd s test we get ) ) l = l B ) ) 2 = B l Because B is bouded With l Hopital s rule we get l the series = diverges. B l l 0 as teds to ifiity) d therefore A example. We wt to determie the character of series 2 )!! = 2)!!. + = 2 )!! 2)!! 2+)!! 2+2)!! = ) ) = Accordig to Raabe s test, the series diverges. = 2 + 6

17 Chapter 3 Joh Carl Friedrich Gauss Bor o 30th April, 777, i Brauschweig, i the Electorate of Bruswick-Lueburg ow part of Lower Saxoy), Germy, J. C. F. Gauss was a Germ mathematici. He is kow as the priceps mathematicorum which mes the price of mathematicis. Ad verily this title suits him up to the hilt. He leart to read d cout by the age of three. At elemetary school whe give a task to sum the itegers from to 00, he produced a correct swer withi secods. His prodigious mid was quickly recogized by his teachers. Supported by his mother d by Duke of Bruswick - Wolfebuttel, who gave him scholarship, Gauss etered the Bruswick Collegium Carolium 792 to 795), d subsequetly he moved to the Uiversity of Gottige795 to 798). While studyig at uiversity, he rediscovered several importt theorems, icludig Bode s law, the biomial theorem, the law of quadratic reciprocity d the prime umber theorem. His breakthrough came i 796, whe he showed that y regular polygo with umber of sides that equals a Fermat prime c be costructed by compass d straightedge. This was the most importt advcemet i this field sice the time of Greek mathematicis. I 799, he proved fudametal theorem of algebra, that every ocostt sigle-variable polyomial over the complex umbers has at least oe root. I 80, he published a book Disquisitioes Arithmeticae Arithmetical Ivestigatios). I the same year, Gauss helped astroomer Giuseppe 7

18 Piazzi with observatio of a small plet Ceres by predictig its positio. This opeed him the doors to astroomy. His brillit work published as Theory of Celestial Movemet remais a corerstoe of astroomical computatio. I 807, he was appoited Professor of Astroomy d Director of the astroomical observatory i Gottige. I 88, while carrig out a geodesic survey of the state of Hover, he iveted a heliotrope, istrumet for measurig positios by reflectig sulight over great distces. He also discovered the possibility of o- Euclide geometries, though he ever published it because of fear of cotroversy. I 828, he published Disquisitioes geerales circa superficies curva, a work o differetial geometry. His famous theorem Theorema Egregium Remarkable Theorem) from this publicatio states i a broad lguage), that the curvature of a surface c be determied by measurig gles d distces o the surface oly. I his late years, Gauss collaborated with physicist professor Wilhelm Weber ivestigatig the theory of terrestrial magetism. By the year 840, he published three importt papers o this subject. Allgemeie Theorie des Erdmagetismus 839) showed that there c be oly two poles i the globe d specified the locatio of the magetic South pole. Together with Weber they discovered Kirchhoff s circuit laws i electricity d costructed electromagetic telegraph, which was able to sed a message over 5000 feet distce. But Gauss was far more iterested i magetic field of Earth. This lead to establishig a world-wide et of magetic observatio poits, foudig of The Magetischer Verei The Magetic Club) d publishig the atlas of geomagetism. He developed a method of measurig the horizotal itesity of the magetic field which was i use for more th oe hudred followig years d worked out the mathematical theory for separatig the ier core d crust) d outer magetospheric) sources of Earth s magetic field. After the year 837, his activity gradually decreased. I 849, he preseted his golde jubilee lecture, 50 years after his diploma. Joh Carl Friedrich Gauss died o 23rd February, 855, i Gottige, Hover ow part of Lower Saxoy, Germy), i his sleep. biography sources from 2d May, 2009): 8

19 3. Gauss test Theorem 3... Let = be a series with positive terms. Assumig that there exist a real umber p, a real umber r > d a real bouded sequece {B } = such that for all = + p + + B r The the series = coverges if d oly if p >. 2 Almost immediately we see that this test is somehow improved versio of Raabe s test. The oly differece is the umber r, which i the former test was explicitly set to 2. So Gauss test is more geeral, allowig us to decide covergece of more series. Not suprisigly the proof is very similar. Proof. If we start with p >, the for sufficietly large we have for egative terms of {B }) ) + or for positive terms of {B }) ) + Sice i both cases = p + B q, q, p) r = p + B r p ) q > + by usig Raabe s test theorem 2..) we c coclude that the series = coverges. As for the other part of equivalece, we will assume that the series = coverges d p =. By usig Bertrd s test theorem 4..) we get l ) ) = l B ) ) r = = B l r l r 2 adopted from o 2d May,

20 With l Hopital s rule we get l 0 as teds to ifiity) d therefore r the series = diverges. A cotradictio. If the series = coverges d p < the for sufficietly large ) ) = p + B + r < d accordig to Raabe s test theorem 2..) the series = diverges. A cotradictio. Thus p >. Theorem Let = be a series with positive terms. Assumig that there exist a real umber p, a real umber r > d a real bouded sequece B such that for all + = + p + B r 3.) The the series = diverges if d oly if p. Proof. Let = be a diverget series d p >. The for sufficietly large ) + = p + B > q, q > r d by Raabe s test theorem 2..) the series = coverges. A cotradictio hece p ). If p = the by usig Bertrd s test theorem 4..) ) ) l = B l + r ) With l Hopital s rule we get l 0 as teds to ifiity) d therefore r the series = diverges. If p < the for sufficietly large ) = p + B + r < We fiish the proof with Raabe s test. Now we show how to make a good use of mysterious lookig sequece B that is, how to reduce our thikig d use algorithmic procedure istead). Look agai at 3.). The best p c be obtaied by a limit ) p = lim )

21 Moreover, we set A = + p Now we try to fid umber r such that 3.3) r >, B = A r < 3.4) That is, {B } is a bouded sequece. If we succeed, we c apply Gauss test. 3 a > 0. A example. We wt to determie the character of the series = + = 2 )!! 2)!! 2+)!! 2+2)!! ) ) a = ) With x ) a = ) a 2 + = + ) p = + p ) x x + o x ) + a + ) 2 we get the fial result ) + a = + o ) ) + + a 2 + ) = a 2 + o) Accordig to Raabe s test, the series coverges whe a > 2 d diverges whe a < 2 but we get o iformatio whe a = 2. We try Gauss test : 2 )!! ) a, 2)!! First, we try to fid p see 3.2)) p = lim 2 )!! 2)!! 2+)!! 2+2)!! 2 = lim 2 ) ) = adopted from o 2d May,

22 Secod, we fid A see 3.3)) A = + p = A = 2 )!! 2)!! 2+)!! 2+2)!! 2 = ) Third, we try to fid r > such that B = A r is bouded see 3.4)) A = If we set r = 2 the Ad because p = the series = = 3 2 ) B = A 2 = < 2 )!! ) 2 2)!! diverges. Therefore, accordig to Gauss d Raabe s test, the series 2 )!! ) a = 2)!! coverges whe a > 2 d diverges whe a 2. 22

23 Chapter 4 Joseph Louis Frcois Bertrd Bor o th March, 822, i Paris, J. L. F Bertrd was a Frech mathematici. He studied at Ecole Polytechique d at the age of 6, he was awarded his first degree. I followig year, he received his doctorate for a thesis o thermodyamics. I 845, Bertrd cojectured that there is at least oe prime betwee d 2 2 for every > 3. The cojecture was later proved by Pafuty Lvovich Chebyshev d it is ow kow as Bertrd s postulate. He made major cotributio to the group theory d published works o differetial geometry, o probability theory Bertrd s paradox ) d o game theory Bertrd paradox). He was also famous for writig textbooks mostly for pupils at secodary schools but later also for more advced studets. I 856, Bertrd was appoited a professor at Ecole Polytechique d also a member of the Paris Academy of Scieces. I 862, he was made a professor at College de Frce. Joseph Louis Frcois Bertrd died o 5th April, 900, i Paris. 2 Bertrd s paradox cocers the probability that arbitrary chord of a circle is loger th a side of equilateral trigle iscribed i the circle. 2 biography sources from 2d May, 2009): Louis Frcois Bertrd 23

24 4. Bertrd s test We begi with the series =2 l ) p p > 0 4.) First we wt to kow for which p the give series coverges. To use itegral criterio theorem.2.), we defie fx) as a fuctio fx) = xl x) p o the iterval, ), which satisfies f) = =, 2, 3,...). The the primitive fuctio F x) is F x) = xl x) p dx = l x l x) p p After some adjustmets, we have the result { l l x if p = F x) = if p p)l x) p l x dx xl x) p+ d from that we c coclude owig to lim x F x)), that the series { coverges if p > l ) p 4.2) diverges if p 0, ] = Now we alyze + : + = l ) p = +)l +)) p = + ) ) l + ) p = + l ) l + l + ) l Usig little-o otatio for asymptotic approximatio whe x teds to ifiity: l + ) = ) x x + o x + ) p = + p x x + o ) x ) p 24

25 We put these two formulas ito + = + ) l + + o )) p = + l = = + ) + + ) + p = + + p l + o )) p l + o = l )) l + o = l ) l d after some adjustmets we get ) ) l = p + o ) + p 4.3) Puttig all this together, we c formulate ew test of covergece, which is kow as Bertrd s test: Theorem 4... Let = be a series with positive terms. If ) ) p >, N N, > N : l p 4.4) + the the series = coverges. If ) ) N N, > N : l 4.5) + the the series = diverges. Proof. Truly, 4.4) implies the existece of two costts q >, ɛ > 0 : q + ɛ = p Therefore we c write for sufficietly large ) ) ) l q + ɛ q + o) + 25

26 with 4.3) q ) l + o = l l ) q +)l +)) q Ad because q > the series = l ) is coverget see 4.2)). Accordig to the secod compariso test theorem.2.9) the series q = coverges as well. Now to prove divergece part. Note that we cot directly use equatio 4.3) here as we did i covergece part) because there c be hidde a egative fuctio f) i o l ), which would make Betrd s test uusable: From 4.3) l +) l +) From 4.5): Because the umbers = + + ) l + o = + l + l f) l +) l +) + l +) l +) + + l l c possibly be foud betwee the umbers ) d + + ) l we do ot have yet eough iformatio to compare + with l +) l +) What we eed is somethig like equatio 4.3) but without the o ) l term. l +) l +) = + ) l + ) l = + )l + l + ) ) = l = + ) l + + ) l + ) l 26 = ) l + l ) =

27 = + + l + ) l + l ) + l = = + + l + ) l + + ) = + l + l + ɛ) l Where ɛ) = + ) l + ) ) 2 + o 0 Oly ow we have eough iformatio to write l + + l + ɛ) l = l +) l +) The secod compariso test implies the divergece of the series = because the series l diverges see 4.2)). Remark The reader should start to suspect that whe we formulated Bertrd s test we commited a small imprecisio. Luckily for us, this imprecisio oly weakeed Bertrd s test but formulatio 4.5) is easy to use). As a result of this flaw, Bertrd s test cot decide covergece of his ow series 4.), as it will be show i Chapter 6. Aother versio of Bertrd s test c be costructed by usig the same Gauss improvemet as with Raabe s test: Theorem Let = be a series with positive terms. Assume that there exist a real umber p, a real umber r > d a real bouded sequece B such that for all + = + + p l + B l ) r The the series = coverges if d oly if p >. 27

28 Proof. Let = be a coverget series d p =. Accordig to the theorem see the Last chapter) l l l + ) ) ) = B l l l l l ) r l ) r usig l Hopital s rule) we get a cotradictio with the covergece we assumed. If p < the for sufficietly large ) ) B l = p + + l ) r < Accordig to the theorem 4.. we get a cotradictio with the covergece we assumed. Thus p >. If p > the = q l, q > with theorem 4.. we c coclude the covergece of series =. Divergece follows by alogy. Theorem Let = be a series with positive terms. Assume that there exist a real umber p, a real umber r > d a real bouded sequece B such that + = + + p l + The the series = diverges if d oly if p. B l ) r 4.6) Proof. Let p =. Accordig to the theorem l l l + + l + B l ) r = B l l l ) r ) 0 ) ) ) = usig l Hopital s rule) we c coclude the divergece of series =. If p < the for sufficietly large l ) + ) = p + B l ) r < 28

29 d the theorem 4.. prooves the divergece of series =. Now let = be a diverget series. The theorem 4.. with p > cotradicts the assumed divergece. Thus p. Remark Agai, we c use B d r to algorithmize the procedure. With p = lim l ) ), A = + + p l we c try to fid such r >, that B = A l ) r will be bouded. If we succeed, we c apply this test. 29

30 Chapter 5 O the uiversal criterio I the chapters 2, 3 d 4 we dealt with various compariso tests but for each oe of them we c costruct a series, that by usig this test we get o iformatio about the character of this series). That is, with tests that combie the secod compariso test theorem.2.9) d some series, whose covergece/divergece we kow. Now we ask whether there c be made some fixed criterio, which will resolve the character of all series with positive terms). Ufortuately, o such criterio c be costructed. The followig two theorems will explai why. Both theorems were adopted from [].) Theorem If = is a coverget series with positive terms the there exists a mootoous sequece {B } = such that lim B = d series = B coverges. Proof. Let = be a coverget series. From the corollary.2.7 we have lim = From the corollary.2.8 we have the umbers ξ, where {ξ } = is icreasig subsequece of atural umbers such that a k < k>ξ Let us costruct the umbers B this way: { 0 if [, ξ ) B = a k if [ξ k, ξ k+ ) 30

31 Now we oly eed to show that = B is coverget: ξ B = 0a k + ξ + a k = k= = k=ξ = Theorem If = is a diverget series with positive terms the there exists a mootoous sequece {B } = such that lim B = 0 d the series = B diverges. Proof. Let = be a diverget series. Here we cosider two cases. First, if lim sup ɛ > 0 the by leavig out all < ɛ 2 we do ot chge the character of series =. We fid all idices such that ɛ 2 d deote them with ξ k. Thus {ξ k } k= is icreasig subsequece of atural umbers. By {B } = we deote such a sequece that { 0 if ξ k for all k B = if = ξ k for some k k Moreover lim B = 0 d B = = ɛ 2 = Secod, if lim = 0 the from the divergece of = we have such icreasig subsequece {ξ } = of atural umbers that ξ + k=ξ a k >, ξ = If we let Ξ = ξ + k=ξ a k 3

32 the the series = Ξ diverges. We costruct the umbers B this way B = k for [ξ k, ξ k+ ), k =, 2,... Now we oly eed to show that the series = B is diverget: B = ξ + a k = Ξ = = = k=ξ = = We showed that to y coverget resp. diverget) series with positive terms we c costruct other positive series b, that coverges resp. diverges) much more slowly. I other words, for y positive coverget resp. diverget) series, there exists some positive series b that coverges resp. diverges) d the secod compariso test theorem.2.9) does ot hold or cot be used). This implies that there c be o ultimate compariso test that is, a test based o some fixed series) as all compariso tests are based o the theorem.2.9. But what we c tell from this is that for y positive series there c be costructed such a criterio based o some kow series b ), which will decide covergece/divergece of series, although fidig such series b might be difficult. Ad that is how Kummer s test works. Agai i plai lguage, for all positive series there exists a compariso test which c decide its character. But there is o such fixed compariso test that c decide the character of all positive series. 32

33 Chapter 6 Erst Eduard Kummer Bor o 29th Juary, 80, i Sorau, Brdeburg the part of Prussia, ow Germy), E. E. Kummer was a Germ mathematici. He etered the Uiversity of Halle with the itetio to study philosophy, but after he wo a prize for his mathematical essay i 83, he deciced for the mathematics. I the same year, he was awarded with a certificate eablig him to teach i schools d, for the stregth of his essay, also a doctorate. For te years he was teachig mathematics d physics at the Gymasium i Liegitz, ow Legica i Pold. I 836, he published a paper o hypergeometric series i Crelle s Joural d this work led Carl G. J. Jacobi together with Peter G. L. Dirichlet to correspod with him. I 839, Kummer was elected to the Berli Academy of Sciece. I 842, he was appoited a professor at the Uiversity of Breslau, ow Wroclaw i Pold, d he beg his reaserch i umber theory. I 855, Kummer became a professor of mathematics at the Uiversity of Berli. With Karl T. W. Weierstrass d Leopold Kroecker, Berli became oe of the leadig mathematical cetres i the world. Kummer helped to advce fuctio theory d umber theory. As for the fuctio theory, he exteded Gauss work o hypergeometric fuctios, givig developmets that are useful i the theory of differetial equatios. I umber theory, he itroduced the cocept of ideal umbers. This work was fudametal to Fermat s last theorem d also to the developmet of 33

34 the rig theory. I 857, he was awarded the Grd Prix by Paris Academy of Scieces for his work o Fermat s last theorem d soo after that, he was give a membership of the Paris Academy of Scieces. I 863, he was elected a Fellow of the Royal Society of Lodo. A year later, he published a work o Kummer surface, a surface that has 6 isolated coical double poits d 6 sigular tget ples. Erst Eduard Kummer died o 4 May, 893, i Berli. biography sources from 2d May, 2009): Kummer

35 6. Kummer s test Here comes probably the most powerful test for covergece, sice it applies to all series with positive terms. 2 : Theorem 6... Let = be a series with positive terms. The the series coverges if d oly if there exist a positive umber A, positive umbers p d umber N N such that for all > N p + p + A 6.) The series diverges if d oly if there exist positive umbers p such that p = d umber N N such that for all > N p + p ) Proof. First we prove the covergece. For the right-to-left implicatio, we adjust the equatio 6.) p p + + A+ With q = p A we c write q q ) Sice we kow the left side of upper iequality, we c costruct a sequece {B } = such that q q + + = B + +, B The sequece {q } = Therefore it has a limit is positive d decreasig follows from 6.3)). Thus the series B = = = 0 lim q < q a q q + + ) = q a lim q > 0 2 adopted from o 2d May,

36 coverges. Ad because B for all B = B I words, the series = B coverges d creates upper boud for the series =, therefore the series = must coverge as well. This is othig more the just a first compariso criteria. We showed that fidig umbers p is equally hard as fidig a coverget series b which creates upper boud for the series i the first compariso test. = For left-to-right implicatio, let p a be a positive umber. Accordig to the theorem 5.0.6, we assume the existece of positive mootoous sequece {B } = Now we shift the idex lim B = 6.4) B = p a + a B = + B + = p a = We defie the sequece {p } = this way where p + + = p + B + lim p = p a lim a k+ B k+ = 0 k= So, usig 6.4) we have for sufficietly large ) Ad fially p p + + = + B + A+ where A > 0 p hece the umbers p are foud. + p + A Remark The reader surely perceives that, i fact, the requiremet 6.4) is ot ecessary as y positive d mootoous sequece {B } with lim B = A is totally sufficiet where A is arbitrary costt). 36

37 For example, let be a coverget series. If we let B = for all d we costruct umbers p usig the terms from p + = p + Kummer s test will cofirm the covergece of series as would first compariso test). Now the divergece part. Proof. For the right-to-left implicatio, we have p = d from 6.2) we get + p p + We c coclude the divergece of series = by usig the secod compariso test theorem.2.9). To prove left-to-right implicatio whe = is diverget), we c assume, accordig to the theorem 5.0.7, the existece of positive d mootoous sequece B such that We have + lim B = 0 6.5) B = = B + B + p = B + Ad after some adjustmets, we get the right formula p It is ot difficult to see, that p the umbers we were lookig for. + p + 0 p p + = d p > 0 for all, thus we foud Remark Agai, the requiremet 6.5) is ot ecessary as y positive d o-icreasig sequece {B } with lim B = A > 0 will do the trick as it is the mootoy we are iterested i). 37

38 As i the previous remark, if is a diverget series d if we let B = for all the Kummer s test will cofirm the divergece. Remark To sum it up, Kummer s test is very powerful because it really works for all the series with positive terms. O the other hd, usig this test is equally difficult as usig the first d secod compariso test. The true stregth of this test therefore lies i the the umbers p. That is, the form of this test is a masterpiece, ot its cotets. To demostrate the power of Kummer s test, we show that Raabe s test d Bertrd s test are i fact its corollaries. As for Raabe s test, if we set p =, we get A > 0, N N, > N : + ) A + ) + A compare with 2.3) + for covergece d ) =, N N, > N : + ) 0 + for divergece. ) compare with 2.4) + We see, that what we c decide with Raabe s test, we c also decide with Kummer s test with p = ) d vice versa, thus they are equivalet. As for Kummer s versio of Bertrd s test, we set p = l d put it ito the left side of 6.): l + ) l + ) = + = l a + ) l + l + )) = + 38

39 = l l l l + ) = l + + ) +) = ) ɛ ) So l a ) ) + ) l + ) = l ɛ ) 6.6) + + Where ɛ) = l + ) +) = + ) )) o 2 = ) 2 +o We go back to 6.). With 6.6), if l the the series = is coverget. A > 0, N N, > N : ) ) + A + ɛ) 6.7) + Compare with Bertrd s test we worked out i the fourth chapter see 4.4)): If A > 0, N N, > N : ) ) l + A 6.8) + the the series = is coverget. Sice ɛ) c get arbitrarily small as teds to ifiity, we c hide it iside the positive costt A. Thus 6.7) d 6.8) are equivalet. Divergece is a bit differet d we will see that i this case, the tests are ot equivalet. That is, Kummer s test is slightly stroger. It is because ow we have zero as a sharp border, while the costt A from the previous case was quite flexible. With 6.2), 6.6) d p = l : if ) l =, N N, > N : 39

40 ) ) l ɛ) + the the series = is diverget. Compare with Bertrd s test see 4.5)): If ) ) N N, > N : l 0 + the the series = is diverget. test Now let s cosider the series =, where = l. Usig Bertrd s + ) l + ) l l +) l + ) l l = +) l Usig Kummer s test with p = l ) ) ) 0 + ) = ɛ) 0 false + ) l + ) l + ) l + ) 0 l 0 0 true There is ifiite umber of series that c be decided oly with Kummer s versio of Bertrd s test, but if we use limits, the tests are equivalet. 40

41 Chapter 7 The Last chapter I this chapter we will retur to Raabe s test to show how we got the sequece l ) i Bertrd s test d cosequetly, how to create limitless p umber of more d more powerful tests. We start by alyzig whe Raabe s test fails. If ) = + ξ), lim ξ) = 0+ + the there exists o such costt p from 2.3), or + ξ). This mes, that the give series coverges more slowly th the series, p >. But it does ot me that the series a p diverges. So what c we do? We eed a series whose covergece/divergece is kow, plus the series must coverge more slowly th. A good cdidate is p l ), sice p ɛ > 0, p > 0, 0, > 0 : ɛ < l ) p To prove use l Hopital s rule.) So, i chapter 4, we worked out Bertrd s test from the series l ). But eve this test fails whe p ) l + ) d the situatio repeats itself. Agai, because = + ξ), lim ξ) = 0+ ɛ > 0, p > 0, 0, > 0 : l ) ɛ < l l ) p 4

42 we c formulate ew, more powerful criterio. First, we have a closer look at the series Let fx) be a fuctio l l l ) p, p > 0 fx) = x l xl l x) p o the iterval e, ) which satisfies f) =. To use itegral criteria we fid the primitive fuctio F x) F x) = x l xl l x) p dx = l l x l l x) p p Thus the series F x) = { l l l ) p l l l x if p = p)l l x) p ) if p { coverges if p > l l x l l x) p+) l x diverges if p 0, ] x dx 7.) Expressig + l l l ) p +) l +)l l +)) p = + ) l + )l l + ))p l l l ) p = after some adjustmets very similar to 4.3)) = + + l + p l l l + o ) l l l 7.2) Now we c create ew test: Theorem Let = be a series with positive terms. If p >, N N, > N : ) ) ) l l l p 7.3) + 42

43 the the series = coverges. If l l l the the series = diverges. N N, > N : ) ) + ) 7.4) Proof. Followig 7.3) p > q > ɛ > 0 p = q + ɛ Thus, for sufficietly large l l l + d with 7.2) l + Because q > the series + ) ) ) q + ɛ q + o) q l l l + o l l l ) q +) l +)l l +)) q l l l ) q l l l coverges see 7.)). Accordig to the secod compariso test theorem.2.9) the series coverges as well. ) As for the divergece, we bump here ito the same problem as we did i Chapter 4 whe we were provig Bertrd s test theorem 4..). We will ot poit out the problem agai.) l l l +) l +) l l +) = + ) l + ) l l + ) l l l = = + ) l + ) l l + l + l l l )) = 43

44 )) + ) l + ) l l + l + l + ) l = = l l l = + ) l + ) l + ) + ) l + ) l + l + ) l l l l = = + + l + l l l + ς) l l l Where: ς) = l l + ) l + ) ) + + ) l + ) l + l + )) l With lim l l + ) l + ) ) = 0 we c simplify ς) ς) + ) l + ) l + l + )) = l = + ) l + l + )) l + l + )) l Ad pull out the biggest part Adjustig 7.4) we get ς) + ) l l + l + l )) l + l l l 0 44

45 Note that the terms we eglected i ς) were all positive so the followig iequality is correct l + l l l Ad fially + + l + l l l + ς) l l l + l l l +) l +) l l +) The secod compariso test implies the divergece of the series because the series diverges see 7.)). l l l Now we state oe of the most geeral versio of Bertrd s test. Lemma Let us deote k-th k N) iteratio of k 0 = ) with l k so l 3 = l l l ). The the series = l k = l l 2... l k l k ) p p > 0 k N coverges for p > d diverges for p. Proof. Let k be arbitrary atural umber. Let fx) be a fuctio o the iterval [, ) such that f) = l l 2... l k l k ) p To use itegral criteria we fid F x) F x) = x l x l 2 x... l k xl k x) dx = p = lk x l k x) p p l k x l k x) p+ l k x...l x)x dx 45

46 { l k x if p = F x) = if p p)l k x) p Now we examie lim x F x) d we c coclude that the series { coverges if p > = l l 2... l k l k ) p diverges if p 0, ] Lemma Usig the otatio from the previous lemma ɛ > 0, p N, k N, 0 : > 0 l k ) ɛ < l k+ ) p Proof. Use substitutio m = l k d l Hopital s rule. Lemma where + ) l + ) l 2 + )... l k + ) l l 2... l k = = + + l l...l k + ϑ) l...l k ϑ) 0 d lim ϑ) = 0 Theorem Let = be a series with positive memebers. If k N, p >, N N, > N : l k l k...l ) ) ) )... p + the the series = coverges. If l k l k the the series = diverges. k N, N N, > N :...l ) ) )... + ) Agai, the theorem c be made ito equivalece. We will use k l i = l )l 2 )...l k ) i= 46

47 Theorem Let = be a series with positive memebers. Assume that there exist atural umber k, a real umber p, umber r > d a real bouded sequece B d = + k + + j= j i= li + p k i= li + B k i= li ) l k ) r The the series = coverges if d oly if p >. Ad the series = diverges if d oly if p. Eve though it might seem that this is a powerful tool to test covergece, it actually c resolve oly a fractio of all possible series with positive terms). We will discuss the results i the chapter amed Epilogue. A example of the case whe the upper criterio fails. Let { } = be a mootoous positive sequece d = be a coverget series. Let {b } = be a sequece such that b = = a2, a N b = 0 for all other It is easy to see that the series = b coverges moreover, if we look at the idex d the term, it coverges very slowly). Now we use these two series to create ew series = Ξ : Ξ = b = 0 Ξ = b b 0 47

48 {Ξ } = : Obviously, the series = Ξ coverges. Now let s try to fid such least mootoous sequece {Υ } = that : Ξ Υ 48

49 It is ot very difficult to see that the sequece we are lookig for is Υ k = 2 k ) 2, 2 ], N Moreover, there are exactly 2 ) terms of value 2 i the sequece {Υ k } k=. Therefore k= Υ k = = 2 2 = = 7.5) Sequece {Υ } = is the least mootoous boud of sequece {Ξ } = d because the coverget series from the theorem is also mootoous, it cot be upper boud for this sequece otherwise it would boud a diverget series from 7.5)). 49

50 Chapter 8 Epilogue Ad that s all folks. We hope that this work has helped all readers to familiarize with the techiques used to determie covergece/divergece of ifiite series with positive terms oly, though). But also to realize, that these special tests Raabe, Gauss, Bertrd) will work oly for a small fractio of them d that o give series c lead to a uiversal compariso test. The example from the previous chapter c also serve as a proof that o mootoous series c lead to a test that will be able to decide all positive series. We are sure that the reader c see where the problem arises. Note that we are distiguishig betwee the terms uiversal criterio d geeral criterio. The former mes a test, based o a fixed series, that c uravel the character of all series with positive terms. That is, the usage of such criterio is straithforward. The latter mes a test e.g. first or secod compariso test) that works for all positive series, but it usually requires additioal d ofte big) effort to make a good use of it as a price for its geerality). Ad that is the situatio with Kummer s test, as this is a geeral test d it truly works for all the series with positive terms) there are. However, whe looked at more closely, we c see that it is just the first d secod compariso criterio i a smart disguise. So makig this test work fidig the umbers p ) c be very difficult. O the other hd, the beefit of this test is that we c see my special criterios as its corollaries. That is, the treasure is the treasure chest itself, ot its cotets, d this is probably the best result we c expect to get with geeral criterios. 50

51 Last but ot least, thk you for readig this work. We hope that someday, somewhere, it will come i hd. 5

52 Refereces to images Joseph Ludwig Raabe Ludwig Raabe.jpeg Joh Carl Friedrich Gauss Friedrich Gauss.jpg Joseph Louis Frcois Bertrd Erst Eduard Kummer 0 all images were retrieved o 2d May,

53 Refereces to biographies Joseph Ludwig Raabe Ludwig Raabe Joh Carl Friedrich Gauss Joseph Louis Frcois Bertrd Louis Frcois Bertrd Erst Eduard Kummer Kummer all texts were retrieved o 2d May,

54 Refereces to other sources of exhaustio test for covergece series 0 all texts were retrieved o 2d May,

55 Bibliography [] Zbyěk Kubáček d Já Valášek. Cvičeia z matematickej alýzy II. Uiverzita Komeského, 996. [2] Tibor Neubru d Jozef Vecko. Matematická alýza II. Uiverzita Komeského,