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Re: Why won't this sum evaluate?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg120609] Re: Why won't this sum evaluate?
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sun, 31 Jul 2011 07:24:23 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <j0oopr$kse$1@smc.vnet.net> <201107301001.GAA25384@smc.vnet.net>

For what it's worth: this sum does not evaluate if one begins with any 
value other than 1, e.g.:


Sum[c^n/(1 + c^(2*n)), {n, 3, Infinity}]

Sum[c^n/(c^(2*n) + 1), {n, 3, Infinity}]

Secondly,  Mathematica can't evaluate the sum from i when c is given a 
specific value, e.g.

With[{c = 1/3}, Sum[c^n/(1 + c^(2*n)), {n, 1, Infinity}]]
Sum[1/(3^n*(3^(-2*n) + 1)), {n, 1, Infinity}]

NSum gets the answer quickly:

With[{c = 1/3}, NSum[c^n/(1 + c^(2*n)), {n, 1, Infinity}]]

0.465259

This agrees with

Sum[c^n/(1 + c^(2*n)), {n, 1, Infinity}] /. c -> 1/3

(1/4)*EllipticTheta[3, 0, 1/3]^2 - 1/4

N[%]

0.46525896368870245

N[%]

0.465259

Andrzej Kozlowski


On 30 Jul 2011, at 12:01, Gary Wardall wrote:

> On Jul 27, 5:20 am, PAR123 <reiser.p... at gmail.com> wrote:
>> In[120]:= $Version
>> Out[120]= "7.0 for Mac OS X x86 (32-bit) (January 30, 2009)"
>>
>> In[122]:= Sum[c^n/(1 + c^(2*n)), {n, 1, Infinity}]
>> Out[122]= -(1/4) + 1/4 EllipticTheta[3, 0, c]^2
>>
>> In[123]:= Sum[c^n/(1 + c^(2*n)), {n, 0, Infinity}]
>> Out[123]= (won't simplify)
>>
>> The only thing different in the two sums is that the second sum is 
from 0 to Infinity rather than 1 to Infinity. Clearly, the n=zero term is 1/2.
>>
>> I have tried various Regularizations and Methods, (not exhaustively) but none seem to work on either of the sums, much less the last.
>>
>> A side problem - Is there a way to determine what Regularization and Method were used when none were specified?
>>
>> Thanks
>
>
> I'm using the latest Mac version 8 and I get the same results as you
> do.
>
> Gary Wardall
>





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