From: math@che.freesurf.fr (Math)
Subject: Re: Approximations of power series
Date: Sat, 25 Sep 1999 18:47:44 GMT
Newsgroups: sci.math
Keywords: Stieltjes' expansion of zeta(s) near s=1

In article <gerry-1509991504240001@abinitio.mpce.mq.edu.au>,
Gerry Myerson wrote:
>In article <37decd15.6214750@news.total.net>, nu__@hotmail.com wrote:
>
>=> A = 1 + (2^n+3^n)/(6^n-4^n)
>=> 
>=> If the sum was of the closed form f(p)=(AP+B)/(P+C) then Sum[k=1..inf]
>=> (1/k)^n should be lim p->inf of f(p) = A.
>=> 
>=> Does anyone have any feedback as to the validity of this approximation 
>=> (if it is indeed one) given above. Thanx... 
>
>Well, as n increases, your A is 1 + (1/2)^n + smaller stuff, and your 
>function, usually called zeta(n), is 1 + (1/2)^n + smaller stuff, so 
>it's not surprising you get some kind of an approximation. For better 
>approximations, see any analytic number theory text, or maybe even 
>any set of math tables that includes the zeta function. 
>

Well there is the well known formula of Stieltjes:

Zeta[ 1+x] = 1/x + gamma + sum( (-1)^n (stieltjes(n) x^n/n!, n=1->oo)

where stieltjes(n) is a family of constants for which there are
several representations and gamma is Euler gamma constant.

The first values are
1->  -0.072815...
2->  -0.0096903..
3->  0.00205383..


Ogg