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.