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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>