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 >

**References**:**Re: Why won't this sum evaluate?***From:*Gary Wardall <gwardall@gmail.com>