# Most of us are aware of the classic Gaussian Integral <msubsup> &#x222B;<!-- ∫ --> 0

Most of us are aware of the classic Gaussian Integral
${\int }_{0}^{\mathrm{\infty }}{e}^{-{x}^{2}}\phantom{\rule{thinmathspace}{0ex}}dx=\frac{\sqrt{\pi }}{2}$
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Belen Bentley
As mentioned in the comments, this sum is related to one of the Jacobi theta functions.
$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{e}^{-{n}^{2}}& =& \frac{1}{2}\left(1+\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\left(\frac{1}{e}\right)}^{{n}^{2}}\right)\\ & =& \frac{1}{2}\left[1+{\vartheta }_{3}\left(0,\frac{1}{e}\right)\right]\\ & \simeq & 1.386\end{array}$
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