How can I compute the following series:
Module[{n = 10^6},
Sqrt[n] * (4^-n) * NSum[Binomial[2*n - 1, n - k]/((2 k - 1)^2 + Pi^2), {k, 1, n}]]
including the cases where n=10^7 and n=10^8?
While trying to compute the first variant, I get the following message from Mathematica 9:
"SequenceLimit::seqlim: The general form of the sequence could not be determined, and the result may be incorrect."
What's the exact version of your Mathematica? On v9.0.1, Win 64bit, I can't reproduce the warning with n=10^6 or n=10^7 or n=10^8. According to this answer, SequenceLimit::seqlim is a evidence that NSum has chosen method "WynnEpsilon", but according to the timing and memory usage, NSum probably choose "EulerMaclaurin" when solving your specific problem. Anyway, I managed to get the warning by raising n to 10^10 and explicitly setting Method to "WynnEpsilon":
$Version
Module[{n = 10^10},
Sqrt[n]*(4^-n)*
NSum[Binomial[2*n - 1, n - k]/((2 k - 1)^2 + Pi^2), {k, 1, n},
Method -> "WynnEpsilon"]] // AbsoluteTiming
And the warning will disappear if I set a higher WorkingPrecision:
Module[{n = 10^10},
Sqrt[n]*(4^-n)*
NSum[Binomial[2*n - 1, n - k]/((2 k - 1)^2 + Pi^2), {k, 1, n},
Method -> "WynnEpsilon", WorkingPrecision -> 30]] // AbsoluteTiming
Not an answer to your original question regarding the proper use of NSum, but I'd like to point out that Mathematica can return a symbolic result in this case for arbitrary $n$:
Sqrt[n]/4^n Sum[Binomial[2*n - 1, n - k]/((2 k - 1)^2 + Pi^2), {k, 1, n}]
which produces
$$\frac{2^{-2 n-1} \sqrt{n} \binom{2 n-1}{n-1} \left((\pi -i) \, _3F_2\left(1,1-n,\frac{1}{2}-\frac{i \pi }{2};n+1,\frac{3}{2}-\frac{i \pi }{2};-1\right)+(\pi +i) \, _3F_2\left(1,1-n,\frac{1}{2}+\frac{i \pi }{2};n+1,\frac{3}{2}+\frac{i \pi }{2};-1\right)\right)}{\pi +\pi ^3}.$$
Unfortunately, evaluation of HypergeometricPFQ is very slow at large values of n, so this is not too helpful.
