TITLE: Evaluation of $\sum_{x=0}^\infty e^{-x^2}$
QUESTION [21 upvotes]: Most of us are aware of the classic Gaussian Integral 
$$\int_0^\infty e^{-x^2}\, dx=\frac{\sqrt{\pi}}{2}$$
I would be interested in evaluating the similar sum
$$\sum_{x=0}^\infty e^{-x^2}$$
Now, because $\exp(-\lfloor x \rfloor^2) \ge \exp(-x)$, we find 
$$\sum_{x=0}^\infty e^{-x^2}= \int_0^\infty e^{-\lfloor x \rfloor^2}\, dx \ge \int_0^\infty e^{-x^2}\, dx=\frac{\sqrt{\pi}}{2}$$
Does a closed form for this sum exist?  If so, what would it be?  I would be very interested in how a closed form would be found for this function.

REPLY [14 votes]: As mentioned in the comments, this sum is related to one of the Jacobi theta functions.
$$\begin{eqnarray*}
\sum_{n=0}^\infty e^{-n^2} 
&=& \frac{1}{2}\left(1 + \sum_{n=-\infty}^\infty \left(\frac{1}{e}\right)^{n^2}\right) \\
&=& \frac{1}{2}\left[1+\vartheta_3\left(0,\frac{1}{e}\right)\right] \\
&\simeq& 1.386
\end{eqnarray*}$$