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Re: Why won't this sum evaluate?

  • To: mathgroup at
  • Subject: [mg120655] Re: Why won't this sum evaluate?
  • From: Dana DeLouis <dana01 at>
  • Date: Tue, 2 Aug 2011 07:13:00 -0400 (EDT)
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> In[122]:= Sum[c^n/(1 + c^(2*n)), {n, 1, Infinity}]
> Out[122]= -(1/4) + 1/4 EllipticTheta[3, 0, c]^2

Hi.  I'll take a crack at it.
I'm not familiar with EllipticTheta, but I do notice what appears to be another issue.

If we look at your solution at values of c=1, and just to either side, we get the following:

f[c_]:=-(1/4)+1/4 EllipticTheta[3,0,c]^2

784.7554\[VeryThinSpace]-9.953975*10^-14 I

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

-(1/4)+1/4 EllipticTheta[3,0,1.001]^2

Notice that it is not evaluated at values of 1.0 or greater.
If we plot the equation at sums of 100, and 1000, we see that there is a spike at 1.
So, the value should be infinity at 1.0 when we sum to infinity.

equ = Sum[c^n/(1+c^(2*n)),{n,1,100}];

equ = Sum[c^n/(1 + c^(2*n)), {n, 1, 1000}];
Plot[equ, {c, -2, 2}, PlotRange -> All]

The function SumConvergence also suggests that the value of c can not be 1

SumConvergence[c^n/(1+c^(2*n)),n] //FullSimplify

I think the program can not handle this sum. 
Sometimes, a workaround is to use a symbolic upper limit instead of infinity, and then change it later.
For example, use z as the upper limit here...

Sum[c^n/(1+c^(2*n)),{n,1,z},Assumptions->Abs[c]!=1] //FullSimplify ;

Sometimes, using Limit as z -> infinity works, but it does not here.  Instead, I just substituted...
%/.z->\[Infinity]  //FullSimplify

Which gives this equation instead:
g[c_]:=(I (QPolyGamma[0,1-(I \[Pi])/(2 Log[c]),c]-QPolyGamma[0,1+(I\[Pi])/(2 Log[c]),c]))/(2 Log[c])

This gives a complex number, with the imaginary part very small.  Just use Chop.
Notice that this does give indeterminate at 1, and a value when c > 1

784.7554\[VeryThinSpace]+0. I


2357.122\[VeryThinSpace]+0. I

If we go back to your problem, and sum from zero...

Sum[c^n/(1+c^(2*n)),{n,0,z},Assumptions->c!=1] //FullSimplify ;

%/.z->\[Infinity]  //FullSimplify

Which leads to this equation:

g[c_]:=(I (QPolyGamma[0,-((I \[Pi])/(2 Log[c])),c]-QPolyGamma[0,(I\[Pi])/(2 Log[c]),c]))/(2 Log[c])

785.2554\[VeryThinSpace]+0. I

Power::infy: Infinite expression 1/0. encountered. >>

2357.622\[VeryThinSpace]+0. I

Notice that at c = 0.999, the differences between the two values are indeed 0.5 as you pointed out.
Not an expert, but hopefully this will help.  :>)

= = = = = = = = = = = =
Dana DeLouis
"8.0 for Mac OS X x86 (64-bit) (November 6, 2010)=94

On Jul 27, 6:20 am, PAR123 <reiser.p... at> 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

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