Computation of standard series

Question

I am stuck on the computation of the following sum:
$\sum_{k=1}^\infty e^{-n^2}\cos(n)$. Simple tricks fail and also i have no idea how to fit it for Fourier series. Are there any other ways?

Answer 1

I fear that you may write it only as a theta function.

\begin{align} S:=\sum_{n=1}^\infty e^{-n^2}\cos(n)&=-\frac 12+\frac 12\sum_{n=-\infty}^\infty e^{-n^2+in}\\ &=-\frac 12+\frac 12\theta_3\left(\frac 12,e^{-1}\right)\\ \end{align} using the definition of the $\theta_3$ Jacobi theta function (see too MathWorld) : $$\theta_3(z,q)=\sum_{n=-\infty}^\infty q^{n^2}e^{2niz}$$ The theta functional equation allows to find a still faster convergent sum ($64$ digits with only $3$ terms) : $$S=-\frac 12+\sqrt{\pi}\;e^{-\frac 14}\left(\frac 12+\sum_{k=1}^\infty e^{-\pi^2 k^2}\cosh(\pi\;k)\right)$$ and even with one term we have (to $15$ digits) : $$S\approx -\frac 12+\sqrt{\pi}\;e^{-\frac 14}\left(\frac 12+ e^{-\pi^2}\cosh(\pi)\right)$$