SUMMATION OF INFINITE SERIES REVISITED I several aricles over he las decade o his web page we have show how o sum cerai iiie series icludig he geomeric series. We wa here o eed his discussio o he geeral problem o idig he sum o- S A obvious soluio occurs whe =. For his ucio we have- S.. By he raio es, his series will coverge o a iie S whe- lim ha so The sum S is easily esablished by oig ha- S or S S Here S represes he amiliar geomeric series. I ca be used o show ha- 6 8 Wih a lile maipulaio i also ollows ha- 6 6 6 ad- 6 6
We ca choose oher ucios or as log as says less ha uiy. So i =siep-=im{ep[--i]}, we ge, a =/, he ideiy- si ep cosh.996877 Sums o his ype are oe ecouered whe akig iverse iie sie rasorms durig he process o solvig he D Laplace equaio. Aoher ype o iiie series is oud whe perormig a Fourier aalysis o boh coiuous ad discoiuous ucios. To demosrae, we cosider he eve ucio =. O muliplyig his by cos ad he iegraig higs over -<<, we ge- cos d cos si Usig he properies o he sie ad cosie ucios, i allows oe o wrie- [cos[ ] [ ] Ploig his equaliy yields he ollowig-
Noe ha he series oly maches he ucio i he rage -<<+, sice his was he choice we made i evaluaig he series. A =, his Fourier epasio yields he impora resul- 8 7 From i also ollows Euler s amous resul ha- S 8 6 Fourier series i geeral are slowly coverge series ad hereore are o lile use or deermiig he values o cerai cosas such as. Aoher ype o iiie series is oud by epadig abou a secod poi =. Oe has- c c c c O diereiaig his epasio imes ad seig =, we ge ha he coeicies have he value c = /!. Hece oe has he Taylor Series-! Oe o he simples orms o his epasio is oud whe =ep. Here all derivaives are equal o. Hece he Taylor series reads- ep ep A = ad =, his says- ep e!!!!.788889 To speed up he covergece rae we ca ake ep = so ha =l. This produces-
[ l] e!!!! l.98689! wih Takig jus he irs ive erms as show i he curly bracke, oe ges he approimaio e.7886 which is already accurae o ive decimal places. Takig he sadard epasio or e up hrough! produces oly a oe decimal place accuracy. As aoher Taylor series epasio cosider he polyomial P give as- P Wha is he ucioal orm o P? here we id ha P =! so ha P=/- provided <. Tha is, we recover he geomeric series- P The polyomial has a irs order pole a = bu reais a iie values away rom his poi. For eample, a =-/ we ge- Here is a plo o P=/- versus over he rage -<<+-
You will oe ha he Taylor epasio abou = o he righ o he sigulariy looks compleely diere rom he epasio o he le. I coverges or <<. I geeral, iiie erm Taylor epasios are valid oly i he regio bewee wo eighborig sigular pois. Epasios direcly over a sigular poi are achievable wih Laure epasios which coai some erms i reciprocal powers o. Oe way o icrease he covergece rae o Taylor epasios is o elescope eighborig erms. Le us demosrae his or he ucio- arca d 7 7 This series is coverge or all iie bu does so very slowly whe ges much above uiy. A =, oe has he amiliar Gregory Formula- 7 Takig he irs oe hudred erm i his series produces oly a wo decimal poi accuracy i /. Ca his covergece rae be icreased? The aswer is yes. Oe o he simples ways o do so is achieved by combiig erms elescopig. Le us demosrae his. Rewrie he series as-
.7898 { } { } { } { }. 7 9 99 9 76 88...789.8 9.7679 So he irs wo erms i he elescoped series yield / o oe place digi accuracy. A aleraive way o icrease he covergece rae or araca series or is o use a wo ucio approach as irs proposed by Joh Machi 68-7. He showed ha- 6arca arca 9 ad used i o calculae o place accuracy. Sice ha ime may oher muliple arca ormulas have bee oud icludig our ow- arca 8 arca 7 7arca 9 arca 68 We have used his las our erm arca ormula o calculae o, decimal places usig our PC. The rapid covergece achieved sems rom he ac ha /N, wih N=8, 7, 9, ad 68, are all very small quaiies. Also he all posiive sigs guaraee ha subseque seps i he series produce values which always will bracke he rue value. To speed up he covergece rae o he sadard iiie erm Taylor Series or ep oe ca elescope higs ad wrie- e!! 6 8! 7 Evaluaig he las series oly up o = yields he digi accurae resul- ep=.788889687766977779699997 I memorize he irs digis o his cosa by he ollowig memoic which I came up wih several years ago. I reads-
.7+AdrewJackso wice+righ riagle+fiboacci hree-ull circle-oe year beore crash-boig je-ed o plaque i Europe-amous roue wes Tha is.7-88-88-9--6-8-77--66 I is also possible o geerae iiie series o he Taylor ype by solvig cerai diereial equaios. Take, or eample, he secod order diereial equaio- O ryig- y +y +y= subjec o y= ad y = y c oe obais he recurrece relaio- c c From his ollows he soluio- / y! /! /! 6 /! Oe recogizes ha he above diereial equaio is he Bessel Equaio or =. So ha he iiie sum represes he zeroh order Bessel ucio J. Is graph looks like his-
A = i leads o he ieresig resul ha- J!!! =.89779 The mius oe erm i he iiie series guaraees ha J is a oscillaory ucio. Is irs zero lies a =.8 as ca readily be esablished by lookig or a sig chage i he series ear =.. Mos iiie series ca be boh diereiaed ad iegraed, allowig oe o geerae addiioal equaliies. Cosider he ollowig eamples- d[arca ] d l si[ ] From hem oe arrives a-
7 si7 si si si l 6 Fially le us look a some equaliies bewee iegrals ad iiie series. Oe ice way o id such relaios is o sar wih he deiiio o a Laplace rasorm by replacig s wih. This yields- Laplace[]= ep F d I ow =si, he F=/{a [+/a ]}. Bu we recogize rom earlier discussios above ha provided a a As a resul we have- a a d a ep si A a=, his yields he ideiy- ep si d O diereiaig m imes wih respec o, oe also ids- m m m m a d a!! ep si
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