THE CLOSED FORMS OF CONVERGENT INFINITE SERIES ESTIMATION OF THE SERIES SUM OF NON-CLOSED FORM ALTERNATING SERIES TO A HIGH DEGREE OF PRECISION.

Transcription

1 THE CLSED FRMS F CNERGENT INFINITE SERIES ESTIMATIN F THE SERIES SUM F NN-CLSED FRM ALTERNATING SERIES T A HIGH DEGREE F PRECISIN. Peter G.Bass. PGBass M er May 0

2 Abstract This paper provies a uerical etho for estiatig the value of the series su of ay alteratig ifiite series, for ay value of the expoet, for which a close for oes ot exist. I particular, it covers the series sus of the opposig expoet Zeta a Eta series, (o expoets), a the Xi series (eve expoets). PGBass i M er..0.0.

3 Cotets..0 Itrouctio..0 The Difficulties Associate with the Solutio of the Close Fors of ζ(), η() a ξ() for the pposig Expoet..0 Derivatio of a Nuerical Algorith to Estiate the Close For alues of the pposig Zeta, Eta a Xi Series et al, to a High Degree of Precisio.. The Algorith Derivatio.. Liitatios..0 Coclusios. APPENDICES. A. Applicatio Exaples for ζ(), η(),, a ξ(), Eve, Plus ther Series. REFERENCES. PGBass ii M er..0.0.

4 .0 Itrouctio. I 70 Leohar Euler evelope his first of three solutios to the Basel proble, the close for of ζ(). His first was the ost prouctive as it also provie close for solutios for ζ() a η() for all eve, a ξ() for all o. However, oe of his ethos provie close for solutios for ay of the above series for the opposig values of. I the iterveig 80 plus years, espite the attetio of ay proficiet atheaticias, this proble has reaie usolve. A brief resue of the ifficulty associate with it is give i the ext Sectio. Cosequetly, it is cosiere that the ext best alterative, is to provie a eas to estiate these close for values to as high a egree of precisio as possible with the iiu aout of coputatio..0 The Difficulties Associate with the Solutio of the Close Fors of ζ(), η() a ξ() for pposig. The close fors of ζ() a η() for eve a ξ() for o, are all irect fuctio of π a are therefore aeable to erivatio via aalysis of suitable circular fuctios. This was Euler's approach i his three ethos, a i the extesio of his first two ethos i [] a [], plus the ethos escribe i [] a []. However, these series with opposig values of are ot irect fuctios of π, a are therefore ot aeable to erivatio via aalysis of circular fuctios. As a exaple, cosier η(). If η() were to be evaluate via the etho of [], the the followig relatioship woul ee to exist F cosθ cosθ cosθ ( θ) cosθ L Differetiatig (.) twice to obtai the Base equatio, (as escribe i [] ), woul give F θ ( θ) f cosθ ( θ) cosθ L cosθ cosθ (.) (.) But, puttig θ 0 i (.) gives η() o the right ha sie, the close for of which is well kow to be l(). While it is possible to costruct a logarithic fuctio i place of f(θ) that satisfies (.), it caot be erive via a Fourier series expasio, a caot therefore be itegrate up to give η(). It woul oly satisfy η(). To provie a relatioship that coul theoretically be itegrate, recourse coul be ae to Newto's expasio of l( x), viz x x x l ( x) x L (.) which, as is well kow, by puttig x, gives l() for the close for of η(). Now, to itegrate (.) up to give η(), woul iitially ea evaluatig the itegral ( x) l x x x x x L x (.) to give the close for of η() by agai puttig x. However, the left ha sie of (.) is a polylogarithic itegral which oe ot appear to have a close for solutio. Itegratio by PGBass M er..0.0.

5 covetioal substitutio ethos, ( i.e. x e y ) oly perits a solutio as a ifiite series which coverges ore slowly tha η(). If (.) is itegrate by parts, i the result the itegral itself cacels leavig other ters which o ot provie the require solutio. ther eas of solvig this itegral are ot apparet, a hece the erivatio of the close fors of the opposig Zeta, Eta a Xi series reais at preset uattaiable..0 Derivatio of a Nuerical Algorith to Estiate the Close For alues of the pposig Zeta, Eta a Xi Series et al.. The Algorith Derivatio. For ay ifiite series that oes ot have a close for solutio, to estiate its sue value to ay esire precisio, it is ecessary to su a sufficiet uber of ters util the applicable ecial place stops chagig. For very slowly covergig series this ca ea suig ay thousas of ters. With the coputig powers that are available toay, this woul ot be too ifficult a task. However, it is still useful to reuce the coputig requireets where possible. To that e, the followig process will, i extree cases reuce these requireets by up to 6 ties. If the partial sus of a alteratig ifiite series are plotte i geeral ters, they ca be represete as i Fig.. below v () () Partial Su alues Evelope Curves (-) v () - Fig.. - Partial Sus vs Ter Nubers for a Alteratig Ifiite Series. I Fig.., (±) v, v () is the ( ± )th partial su. are iter-partial su values o the evelope curves. is the series ter eoiator separatio costat. Also ote that PGBass M er..0.0.

6 PGBass M er (.) Where is the series expoet. To begi, eterie a estiate of the series su base upo the o ter partial su. First v (.) So that the estiate is v (.) Which reuces to, via substitutio of ters fro (.) (.) Now eterie a estiate of the series su base upo the eve ter partial su (). First v (.) So that the estiate is v E (.6) Which reuces to, via substitutio of ters fro (.) E (.7) E () will always be greater tha, a for low to eiu values of, E () will be greater tha the close for value, a, lower. Therefore, subtractig (.) fro (.7) yiels ε (.8)

7 a this reuces to ε ( ) ( ) (.9) ε is a easure of the rage with withi which the series su lies. Now (.9) ca easily be solve i ters of whe is very sall, i.e. sigle igit. Eve so, it is still easier to siply plug values of ito (.9) to fi the value of ε esire. The that uber for is use to eterie a E () so givig the partial su rage withi which the series su lies. The series su estiate, to the esire egree of precisio, is the give by the average of a E (). The brief table below gives soe rao figures of iiu expecte precisio for η() usig this etho., ( ) ε 9 9.6E E- 8.E-6 7.0E-9 Table. - Estiate Precisio for arious a for η(). As a specific exaple, to eterie η() to a precisio of <E-8, usig covetioal ethos requires suig 6 ters. As see fro the Table above, the etho here requires oly 9 ters to achieve the sae or better egree of precisio, a reuctio of soe 9. ties. I fact, usig covetio suig ethos to achieve exactly the sae precisio as obtaie here, woul require suig 88 ters. It shoul also be ote that this etho ca be use with ay alteratig coverget ifiite series for which a close for is ot available a for which the series ter eoiator separatio is a costat. Appeix A provies soe exaples for eteriig the series su of η(), (a therefore ζ()), for o, a for ξ() for eve, plus other series.. Liitatios. As state earlier, for low to eiu values of, a E () bracket the close for value of the series uer ivestigatio, a so the precisio of its estiate is at least equal to the value of ε selecte, a i ost cases uch better. However, whe is icrease to high values, the close for value ca fall just outsie of the rage to E (). Whe this happes the precisio of the estiate give by the average of a E () will be less tha the value of ε. The precisio will the reai costat, irrespective of ay icrease value of a associate ε, at the level achieve whe a E () last brackete the close for value. Usig the exaple of η(), the precisio obtaie for 9 is e-0, agaist a selecte ε of E-8. Whe is icrease to 887 to give a ε of E- the close for rops just outsie the associate rage of a E (), a a slightly lower precisio of E- is obtaie. As is icrease further this level of precisio reais substatially costat irrespective of the level of ε specifie by the icreasig value of. The figures i this exaple oly apply to the value of the expoet. Whe is higher tha this value, the liitig level of precisio is well i excess of E-. PGBass M er..0.0.

8 .0 Coclusios. While this etho provies a siple eas of estiatig the series su of applicable alteratig ifiite series, it still requires suig, i soe cases, a cosierable uber of ters of the series. However, i extree cases it ca ea suig over 60 ties less tha the uber of ters sue covetioally to achieve the sae egree of precisio. Also, it provies very quickly, prior to ay large scale coputatio, the uber of ters eee to obtai the esire precisio, so avoiig oitorig a particular ecial place for its costacy i a covetioal suig process as ore ters are ae to the su. The values that ca be calculate usig this etho, beig liite to a certai uber of ecial places is i fact o ifferet to those series for which a close for is available. This is so because i ost cases, such close fors are either fuctios of π or logarithic, both of which beig irratioal ca theselves oly be expresse to a liite uber of ecial places. That is ot to say that the solutios obtaie here are equivalet to those eterie fro a close for. The solutios obtaie here are ot those of a close for, but of a series su. The ai liitatio of the etho, apart fro the ecessity to still su a uber of ters of the series, is that of the bottoig out of the level of precisio that ca be obtaie as the value of is icrease. However, this liitatio is cosiere oly a ior oe as the level of that causes this effect is very high, a therefore eve for the iiu value of, the iiu egree of precisio so obtaie is still extreely high. The lack of solutios of these series i close for oes ot ea that such solutios o ot exist i the for of eleetary fuctios. It is believe that such solutios o exist a as iscusse i the text above, are expecte to be of logarithic or polylogarithic for. Their evaluatio, i cojuctio with the etho of [], appears to epe upo the erivatio of a close for solutio to the itegral epicte by Eq.(.), a this eteriatio ay require the evelopet of soe "ew aths". Fially, although reuat, this etho is of course also applicable to those series for which a close for is available. Appeix A. Applicatio Exaples for η(), ζ(), ( o), a ξ(), ( Eve), Plus ther Applicable Series. (i) Series su estiate of η() a ζ() to a iiu precisio of E-8. I this case. Therefore (.9) becoes ε ( ) ( ) (A.) It is iportat to ote that the values of chose to isert ito (A.) to obtai the require ε ust be values that provie a positive ter i the series. Thus substitutio of trial values of ito (A.) to obtai a value of ε E-8 yiel 9, a so PGBass M er..0.0.

9 (A.) The 9 becoes x9 (A.) Which, upo isertio of (A.) becoes (A.) Siilarly E 0 evaluates to E (A.) a the average of (A.) a (A.) is η ( ) (A.6) a this is withi E-0 of η() calculate to places. ζ() is the calculate as ς ( ) η( ) η( ) (A.7) which is withi E-0 of ζ() calculate to places. (ii) Series su estiate of ξ() to a precisio of E-8. I this case a for a ε of E-8, the value of require fro (.9) is 7 to give To give E ξ ( ) (A.8) (A.9) (A.0) (A.) Which is withi E-0 of ξ() calculate to places. (iii) Series su estiate, to a precisio of E-8, of. S ( ) L 9 7 (A.) PGBass 6 M er..0.0.

10 I this case a the require is to give To give E S ( ) Which is withi -.698E-9 of S() calculate to places. (A.) (A.) (A.) (A.6) (iv) Series su estiate, to a precisio of E-, of. S ( 6) L (A.7) I this case a the require is 6 to give (A.8) (A.9) To give E (A.0) S( 6 ) (A.) Which is withi E- of S(6) calculate to places. Refereces. [] P.G.Bass, The Close For of Coverget Ifiite Series - - A Extesio of Leohar Euler's First Metho, [] P.G.Bass, The Close For of Coverget Ifiite Series - - A Extesio of Leohar Euler's Seco Metho, [] P.G.Bass, The Close For of Coverget Ifiite Series - - Geeralisatio ia Fourier Series Expasio, [] P.G.Bass, The Close For of Coverget Ifiite Series - - Deteriatio ia Recursive Itegratio, PGBass 7 M er..0.0.