From: robjohn9@idt.net (Rob Johnson) Subject: Re: Sum and Product Involving e^x Date: 19 Jan 2000 14:48:10 GMT Newsgroups: sci.math Summary: [missing] In article <863ls9$a9f@mcmail.cis.McMaster.CA>, kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) wrote: >In article , >Leroy Quet wrote: >:Let S(x)= sum_{k=1}^infinity [(1 +x/k)^k -e^x] >:and P(x)= product_{k=1}^infinity [(1 +x/k)^k/ e^x]. >:Do S(x) and P(x) both converge or both diverge? >:If they converge, are there closed functions for what they converge >:to, at least at x=1? >:Thanks, >:Leroy Quet > >They both diverge: the sum to -infinity, the product to 0. Yes, if the >infinite product of nonzero numbers converges to zero, it is called >divergent. > > The "correction" e^x is not strong enough, and the nature of divergence >is about the same as the divergence of the harmonic series. > But here is an infinite product that can be expressed in closed form, as >a non-constant function of x, and I am leaving it as a challenge (although >I suspect it was known to Euler or some of his contemporaries): > >Product for n from 0 to infinity of > > (1/e) * (1 + 1/(x + n))^(x + n + 1/2) k-1 --- n+x+1/2 -1 | | ((n+x+1)/(n+x)) e n=0 k k-1 -k --- n+x-1/2 --- -n-x-1/2 = e | | (n+x) | | (n+x) n=1 n=0 k-1 -k k+x-1/2 -x-1/2 --- = e (k+x) x / | | n+x n=1 -k k+x-1/2 -x-1/2 = e (k+x) x Gamma(1+x) / Gamma(k+x) -k k+x-1/2 -x+1/2 k+x-1/2 -k-x+1 ~ e (k+x) x Gamma(x) / [ sqrt(2 pi) (k+x-1) e ] x-1 k+x-1/2 -x+1/2 = e (1+1/(k+x-1)) x Gamma(x) / sqrt(2 pi) x -x+1/2 -> e x Gamma(x) / sqrt(2 pi) Therefore, oo --- n+x+1/2 -1 x -x+1/2 | | (1+1/(n+x)) e = e x Gamma(x) / sqrt(2 pi) n=0 Rob Johnson robjohn9@idt.net