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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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