From: Raymond Manzoni <raymman@club-internet.fr>
Subject: Re: Limit involving sum
Date: Tue, 27 Jul 1999 19:16:29 +0200
Newsgroups: sci.math
Keywords: Evaluating infinite sums with PSLQ

Leroy Quet wrote:

> I got "closed" results for S(0) through S(4). I got them a while ago,
> and I can?t remember how. As a result, I am not sure of my results?
> accuracy.
> S(0)=1.
> S(1)=zeta(2)=pi^2/6.
> S(2)=3zeta(3).
> S(3)=10zeta(4)=pi^4/9.
> S(4)=30zeta(5)+6zeta(2)zeta(3)=30zeta(5)+pi^2zeta(3).
> (where zeta(s)=sum_{m=1}^infinity [1/m^s].)
> The sums can converge slowly.
> I don't know what S(r)/r! approaches.
> However, I believe it?s likely that, if
> S?(s)=sum_{m=1}^infinity [(H(m))^s/m^2],
> limit, s->infinity, S?(s)/s! >0.
> (S?(s)>sum_{m=1}^infinity [(lnm)^s/m^2].
> But integral_1^infinity [(lnx)^s/x^2]dx=s!)
> Thanks,
> Leroy Quet

Leroy Quet


    Hi,

    I verified your particular values and found there were all right
(easy for 1 or 2 by
telescoping and then numerically using mupad and PSLQ with 40 right
figures with the
help of Ref 1).

    Here are the next values PSLQ found yesterday:
    S(5) = 17/90*PI^6+45/2*zeta(3)^2
    S(6) = 644*zeta(7)+145/6*PI^2*zeta(5)+33/10*PI^4*zeta(3)

    Well, all this is a research topic in itself and your direct methods
are probably more
adapted to your initial quest (which looks numerically right)!

    Sorry since it doesn't really help!

                Raymond Manzoni


    Ref 1: when m*(m+1) is replaced by m^k (with k>1 so that there's no
real intersection
with the previous formulas) you get the Sh functions described here :

Bailey and all : "Experimental Evaluation of Euler Sums" and "Explicit
evaluation of Euler
sums" at http://www.cecm.sfu.ca/preprints/1993pp.html

    Refs 2:PSLQ and Euler sums :
http://www.cecm.sfu.ca/organics/papers/bailey/paper/html/node6.html#SECTION00060000000000000000
==============================================================================
[It is perhaps worth remarking that this program, and the newer  LLL  
 algorithm, require excellent numerical values for the quantities to be
 tested, test only fixed monomials in the several quantities, and only
 compute _apparent_ rational linear combinations among the monomials. --djr]