Summation Method for Some Special Series Exactly

Transcription

1 The Iteratioal Joural of Mathematis, Siee, Tehology ad Maagemet (ISSN : 39-85) Vol. Issue Summatio Method for Some Speial Series Eatly D.A.Gismalla Deptt. Of Mathematis & omputer Studies Faulty of Siee & Tehology Gezira Uiversity, Wadi Medai, Suda dfgismalla@hotmail.om Abstrat-Effiiet Numerial methods for series summatio to a high deimal plaes of auray a be foud elsewhere. However, most of these methods a't sum may types of speial series eatly beause either of roudig's errors or these methods sometimes fail to ompute slowly overget series as i [. I this paper, We shall desribe a approah that a be applied to sum some speial types of series eatly wheever these speial series a be emerged ad suit our riterio method. Our method atually uses two approahes for epressig futios as series of hebyshev polyomials approimatio.the first approah is Taylor's Epasio Series where eah mooomial =0,,,3,. i Taylor's series is replaed ad represeted by its orrespodig hebyshev idetity.the seod approah is heybeshev Approimatio Series for a partiular futio. Depedig o this partiular futio, We ompare the orrespodig oeffiiets assoiated with T j () j=0,,,3,.. betwee the two series. Eah oeffiiet i first approah emerge ad geerate a ifiite series with its sum eatly equals the orrespodig oeffiiet i the seod epasio. The partiular futio that We shall osider to be epressed first i Taylor's Method ad seod i hebyshev Series to emerge the may ifiite series is ½l[.The emerged ifiite may series eah with its eat sum are give by Eq.() for,,3,4,... () where is Biomial offiiet ( )! ad ( )!( )! Alteratively, Eq.() a be rewritte as ( j) 4 ( ) 3 j j ( j ) 3 j for,,3,4,...( ) Eq.() shows ifiite formulae series that represets the reiproal of odd umbers. I. THE VALIDITY OPINION FOR EXAT SUMMATION May ifiite series havig a eat summatio a be foud ad evaluated it. Here, for demostratio,we list two ifiite series,eah with its eat sum that a be evaluated easily. These series are i Eq.(3) ad Eq.(4) (3) ( )( ) 0 We show the sum of the ifiite series i Eq.(3) is eatly equals oe. osider its partial sum S that a be epressed as S (. ).3... ( ) ( )( ) ( )... ( ) ( ) 3 S... 3 Observe that all the terms vaish with eah other eept the first ad the last terms. This implies that S osequetly, s as This shows that the ifiite series i Eq.(3) overges ad its sum equals oe, i.e. S lim S ( )( ) 0 Similarly, We a show, aother ifiite series i Eq.(4) has its sum equals ad it is give by ( ) (4) Page 8

2 The Iteratioal Joural of Mathematis, Siee, Tehology ad Maagemet (ISSN : 39-85) Vol. Issue This demostrates that there are so may ifiite series where eah overges eatly to its sum without the eed for seeig some umerial methods to to ompute it. Suh types of series a be useful for may appliatios. II. THE ORTHOGONALITY AND THE FUNDAMENTAL IDENTITIES FOR HEBYSHEV POLYNOMIALS The hebyshev polyomials of the first id are defied through the idetity T ( ) os( os ) Now substitute = os( ) to (5) get T(os( )) os( ) (6) Hee, the first few hebyshev polyomials of the first id are T o () = T () = T () = T 3 () = T 4 () = T 5 () = T 6 () = The et idetity epressed without proof is the epasio of hebyshev polyomials T () as suessive powers i, where is a positive iteger T () = r j 0 α j -j (7) where the oeffiiets α's are determied by α 0 = - ad α j = (-) j -j - j (8) j j for j = () r ad r = [ is the iteger part of.where j is the j orrespodig Biomial oeffiiet. For a alterative epressio of Eq. (7), see [3. The third well ow idetity that we have rewritte, is the epliit represetatio of i terms of hebyshev polyomial T j (), j=0(), for some positive iteger. That is = ' j T j () (9) j 0 where j = -+ (0) ( j)/ for j =() if is odd & j =0() if is eve. The prime dash i the summatio meas that oeffiiet of T 0 () i (9) should be halved. Further it a be show that hebyshev polyomials are orthogoal polyomials with respet to the weightig futio ad satisfies Eq.() T m ( ) T ( ) d os( m)os( ) d 0 for m 0, 0 m () for m 0 Furthermore, Rodriguez represetatio idetity is give by Eq.() III. ( ) T ( ) d d ( ) * ( [( ) )! () TAYLOR S EXPANSION FOR ½l[ WITH ITS MONONOMIAL REPLAED BY HEBYSHEV IDENTITY. Taylor's epasio as a ifiite series for ay futio y() at the poit 0 = 0 is give by y()= y ( 0) (3) 0! Hee, Taylor's Epasio for ½l[ at the poit 0 = 0 is give by l[ (4) Observe that the power of eah mooomial =,, 3, is odd. Page 9

3 The Iteratioal Joural of Mathematis, Siee, Tehology ad Maagemet (ISSN : 39-85) Vol. Issue This implies that,we eed to epress eah mooomial as hebyshev idetity by Eq.(9) ad Eq.(0). We must tae odd i Eq.(0), i.e. =, 3, 5, -. So, if, We replae eah mooomial by its assoiate hebyshev idetity, Eq.(4) beomes l[ ( - ) T (5) Where is Biomial oeffiiet as i Eq.(0) Now, if We rewrite Eq.(5) as Eq.(6) below l[ T (6) where the oeffiiet for =,,3,4,,is the sum for the ifiite series iside the braet i Eq.(5) whih must be determied. We get the may ifiite series as - for,,3,.. (7) The sum of eah series i Eq.(5) is eatly equals for,,3,4,... as,we will derive i the followig setio that l[ a be epressed as a ifiite Eq.(8) l[ hebyshev series as i T (8) IV. HEBYSHEV EXPANSION SERIES FOR ½l[ To develop the futio f()=½l[(+)/(-) as a series of hebyshev polyomials, We suppose that f() a be epressed as l[ T o (9) Now multiply both sides of Eq.(9) with T () ad itegrate with respet to the weight futio to get by usig the orthogoality proess as i Eq.() ½l[(+ )/( - ) T ( ) d (9) The oeffiiet =0 for =0,,4,., ad this a be see from Eq.(9) that the itegrad i it is a odd futio itegrated o the iterval [-,. This implies that, We oly see to evaluate the odd oeffiiets for =,3,5,., -, Now substitute the Rodriguez represetatio Eq.() ito Eq.(9) ( ) ( ) T ( ) ( )! ad itegrate by parts to get d d ( ) ( ).. [0 ( )! d [ ( ) d d () (0) The reader must observed the limits a't be determied uless, We use the L'Hospital Rule i Eq.(0). Now, if We itegrate by parts while differetiate (-) th times both the itegrads i the braet, We get ( ).. [0 ( )!*{ ( )! ( ) [( ) ( ) ( ) ( ) d} ( ) (0) Hee, the oeffiiets for a odd umbers =,3,5,.., -, a be rewritte as Page 0

4 The Iteratioal Joural of Mathematis, Siee, Tehology ad Maagemet (ISSN : 39-85) Vol. Issue.. [( )![ I I () ( )! Where the itegrals i Eq.() beomes ( ) ( ) I = [ d ad ( ) ( ) I = [ d () (3) Observe that the itegrals I i Eq.() ad I i Eq.(3) otais two equals itegrals give by ( ) [ ( ) d d [ (4) whih are both equal. Trasforms oe of the itegrals i Eq.(4) by substitutig =-y to get the equality.further observes that these two itegrals i Eq.(4) otribute zero through the additio of I & I i Eq.(). Substitute =os ito I = ( ) [ d ad = os ito I = ( ) [ d to get Wale's Formula I = si ( ) d 0 () ( )! I ( ) (!) Now,ombie this i Eq.() to get.. [ ( )!* ( )! ( )! (5) ( ) (!) Put ( )! ( )! ( )! ( ) ( ) ito Eq.(5) to get,,3,.., (6) Sie the oeffiiets 's are zeros whe is eve, the the oeffiiet i Eq.(9) a be writte as,,3,..., (6) Hee the represetatio of the futio f()=½l[(+)/(-) as hebyshev Series a be epressed as ½l[(+)/(-)= T ( ) (7) Where T ( ) is hebyshev Polyomial of degree - Now substitute the oeffiiets i Eq.(7) ito Eq.(6) ad ompare with Eq.(7),to get the required ifiite may series with their eat sum as i Eq.(). for,,3,4,... V. ONLUSION () I a future wor we shall ivestigate whih best umerial methods are available to ompute those ifiite series i Eq.() to a very high auray. It is well ow that as We desribe i [, Levi's Trasform sum some types of ifiite series to a very high auray usig FORTRAN LANGUAGE. However, despite the fat Levi's Trasform a be osidered as oe of those best umerial methods to sum ifiite series, sometimes it has a drawba. It fails to sum some types of a ifiite series ad it a't evaluate overget series eatly as i Eq.() I a future wor, we will write software programs to ompute the sum for ifiite series to a high deimal plaes of auray. I partiular,we will apply the program to ompute the series as type of Eq.(). Page

5 The Iteratioal Joural of Mathematis, Siee, Tehology ad Maagemet (ISSN : 39-85) Vol. Issue REFERENES [ D.A.Gismalla & A.M.ohe Aeleratio of overgee of Series for ertai Multiple Itegrals I.J..M, Vol. 4, pp 55-68, 987. [ Milto Abramowitz & Iree A_Stegu Hadboo of Mathematial Futios Dover publiatios, I., NEW YORK [3 arl Eri Froberg Numerial Mathematis,Theory ad omputer Appliatios.The Bejimi ummig Publishig ompay, I., 985 [4 hebyshev Approimatios for os(½ ח 4 ) ½)is da ח 4 ) eht 0, Proeedig of Iteratioal oferee o omputig De. 8th -9th.00, New-Delhi, INDIA Page