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MathGroup Archive 2000

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Re: Infinite sum of n^2 Exp[-n^2]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg26483] Re: Infinite sum of n^2 Exp[-n^2]
  • From: "Kevin J. McCann" <KevinMcCann at home.com>
  • Date: Thu, 28 Dec 2000 02:52:22 -0500 (EST)
  • References: <9217t2$com@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Sort of.

Try 

f[a_]=Sum[Exp[-a n^2],{n,1,Infinity}]

then your summation is just the derivative with a->1

f'[a]==-Sum[n^2 Exp[-a n^2],{n,1,Infinity}]

Evaluate the derivative WRT a with a=1

-f'[1]
or
=-f'[1]//N

The first is given in terms of the EllipticTheta function, which is closed 
form, but maybe not what you would like.

BTW, you can't go {n,-Infinity,Infinity} - blows up.

Kevin


The wise words of A. E. Siegman on 22 Dec 2000 22:58:26 -0500:

> Mathematica can do the infinite sums (-Infinity to Infinity) of
> 
>    Exp[-n^2]
> 
> and also
> 
>    Exp[- a n^2]
> 
> in closed form, but not
> 
>    n^2 Exp[-n^2]
> 
> or better
> 
>    n^2 Exp[- a n^2]
> 
> Are there known closed analytical forms for these?
> 
> Thanks     siegman@@stanford.edu
> 



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