Summation Method for Some Special Series Exactly

D.A.Gismalla

Efficient Numerical methods for series summation to high decimal places of accuracy can be found elsewhere. However, most of these methods can’t sum many types of special series exactly because either of rounding's errors or these methods sometimes fail to compute slowly convergent series as in [1] . In this paper , We shall describe an approach that can be applied to sum some special types of series exactly whenever these special series can be emerged and suit our criterion method . Our method actually uses two approaches for expressing functions as series of Chebyshev polynomials approximation .The first approach is Taylor's Expansion Series where each mononomial xn n=0,1,2,3,…. in Taylor's series is replaced and represented by its corresponding Chebyshev identity .The second approach is Cheybeshev Approximation Series for a particular function . Depending on this particular function, We compare the corresponding coefficients associated with Tj(x) j=0,1,2,3,….. between the two series. Each coefficient in first approach emerge and generate an infinite series with its sum exactly equals the corresponding coefficient in the second expansion. The particular function that We shall consider to be expressed first in Taylor's Method and second in Chebyshev Series to emerge the many infinite series is ½ln[ x x   1 1 ] .The emerged infinite many series each with its exact sum are given by Eqn.(1) 2 1 2 1 2 1 2 2 2 2 1 1       ï‚¥     n k k n C k n k k for n 1,2,3,4,...... (1) where is Binomial Cofficient k k n C2 1 1    ( )!( 1)! 2 1 (2 1)! 1         k n k n k k k n and C Alternatively, Eqn.(1) can be rewritten as 2( ) 3 2 2 1 2( ) 4 2 2( ) 3 1      ï‚¥       j n j n C j j n j n 1,2,3,4,......(2) 2 1 2    for n n Eqn.(2) shows infinite formulae series that represents the reciprocal of odd numbers.