Following a similar approach it should be possible to extend such summations to the more general function-

Transcription

1 F ( ) EVALUATION OF THE FUNCTION I recet ewspper rticle (St.Petersburg Ties, Sudy Oct.9, 008) there ppered story bout high school studet with theticl skills with iterest i ttedig college s th jor. I oe of the pictures he is show workig out the su of the series- () F by re-groupig procedure bsed o the sipler sus- 0) d ) 0) ) 0) Followig siilr pproch it should be possible to exted such sutios to the ore geerl fuctio- ( ) F Oe would expect the sus to go s- ) Ak k) A0 0) A ) A )... A F ( ) k 0 We hve crried out such evlutios with id of our coputer to fid-

2 ) 0) ) ) 6 ) ) ) 6 ) 5) ) ) 50 5) 7) ) ) 08 6) 85) ) ) ) 966 There ppers to be o obvious ptter i the ubers A k or i the vlues of the sus foud for ) other th tht the vlue of ) icreses with icresig d tht the sus re ll eve itegers. This suggests tht lterte route ust be tke to fid geerl ptter for ) d to fid sipler lgorith for fidig its vlues. Let us do this. Recll fro erlier discussio tht it is possible to covert ifiite series to itegrls vi Lplce trsfors. The procedure sys tht- h( t) H ( ) t 0 exp( t) where H() is the Lplce trsfor of h(t) with replced by s. Usig our MAPLE coputer progr oe c show tht- s ivlplce( ) Heviside( t l()) Dirc(, t l()) s where Dirc(, t-l()) refers to the th derivtive of the Dirc Delt Fuctio. Oe c thus write tht- ) d Dirc( t l()) exp( t) t l( ) Next we ote by itegrtio by prts tht for y cotiuous fuctio f(t) oe hs- t f ( t) Dirc(, t ) ( ) d f ( t) t

3 so tht- ) ( ) d exp( ) t t l() This is extreely siple lgorith for fidig the su of the series ). It leds to the followig results- )! ) []!! 6 ) []!! 6 ) [7]! [6]!! ) [5]! [5]! [0]! 5! ) []! [90]! [65]! [5] 5! 6! 966 This tie oe otices defiite ptter. The lst ter i ech of the sus equls!, d the secod to the lst s -[(-)/](-) -. The ters further to the left becoe ore cubersoe before they gi siplify yieldig! (-) ( (-) -) d! (-)() for the first two ters to the right of the equlity sigs. I geerl oe hs- ) ( )! ( )( 5) 96 ( )( ) 8 O( ) So for it follows tht- F ()!! []! [] 6 d for we hve- F ()! [ (7) 96 () ] 8 Oe c lso costruct Pscl like trigle s show- 50

4 !!!! 5!5 6! 6 7! 7 8! 8 9! 9 SUM ) ) - 6 ) - 6 ) ) ) ) ) ) d the c red the results for vrious )s directly. Tke the cse 8. It yields- 8) []! [7]! 6 [66] 6! 7 [8] 7! [966]! [70]! [] 8! 8,09,670 5 [050] 5! To deostrte the power of the bove derivtive lgorith for lrger vlue of, look t the cse ). It tkes oly frctio of secod to yield the followig digit result- ) d [ ] t e t l() Note gi tht the su is eve uber. Filly let us briefly etio how it is possible to obti pproxite vlues for ifiite series such s the preset oe. Oe eeds oly to replce the iteger vlues i the ifiite su by the vrible x d the write- F ( ) x dx x exp[ xl()] dx Γ( ) /[l()] x x 0 x 0 where the lst itegrl hs bee evluted exctly for this cse by recogizig tht it is just the Lplce trsfor of x with s replced by l(). To see how good this pproxitio is, we fid, for exple, tht Γ()/[l()] d

5 Γ(7)/[l()] These vlues should be copred to the exct sus )6 d 6)966 give bove. Aother dvtge of such itegrl pproxitios is tht oe c see t wht pproxite vlue the ters i the series rech their xiu. I the preset cse siple differetitio x / x shows tht the xiu vlue of the ters i the preset series occurs er x/l() d hs pproxite vlue of V [/l()] / [/l()]. For 6 the xiu occurs er d hs vlue V0.768 s is icely cofired by the followig grph- Oct., 008 Note dded Jue, 009 I recetly received eil fro Michel Rode(the studet writte up i the the St.Petersburg Ties Oct.008) who tells e he will be ttedig the Uiversity of Florid strtig this fll s th jor. He is presetly workig o the geerlized su- b b b b vi his regroupig pproch. Oe c lso crry out the sutio usig the bove derived differetitio pproch bsed o Lplce trsfors. Oe fids tht-

6 d S(, b) ( ) b dx exp( x) x l( b) which produces the solutio- S (, b) [( ) K(, ) b ( b ) ] ( b ) where K(,) re coefficiets risig i crryig out the idicted differetitios. The vlues of these coefficiets c best be preseted i tble for s follows- K(,) K(,) K(,) K(,) K(,5) K(,6) !!(-)/ Thus, for exple, S (,) ( )( ) 0 ( ) ( )( ) 6 d S (,5) 5 ( 0 ) 6( ) ( 5 ) ( ) 85 8