Re: Here is a mathematica challenge for fun

*To*: mathgroup at smc.vnet.net*Subject*: [mg28765] Re: [mg28670] Here is a mathematica challenge for fun*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Fri, 11 May 2001 20:00:43 -0400 (EDT)*References*: <200105060511.BAA27174@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Singularity Chaser wrote: > > Use Mathematica (or other methods) to find the asymptotic expansion of > the following Sum in the 0+ limit > > Ksi(x)=Sum[1/n^(n*x),{n,1,Infinity}] > [...] Asymptotically it behaves as Exp[ProductLog[1/x]. More specifically, it can be written as (1+f[x])*Exp[ProductLog[1/x]] where f[x]->0 as x->0. I think f[x] is (at least eventually) negative, but am unable to pin it down closer than that. To give an idea of the accuracy of this approximation, for x=1/10000 we have a sum around 1326 whereas Exp[ProductLog[10000]] is around 1383. It would be useful to know why this limiting behavior is of interest. Daniel Lichtblau Wolfram Research

**References**:**Here is a mathematica challenge for fun***From:*Singularity Chaser <sotirisgk1@aol.com>