From: Raymond Manzoni 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]