This problem was inspired by a typo on a homework assignment for Calculus 2, which covers integration and series.
Find the sum of $$\sum_{n=0}^{\infty} \frac{1}{2^{n^2}}$$
Does anyone have any idea regarding where they'd begin with this problem? Maybe some fancy residue calculus trick?
Sums of this form are called Jacobi $\theta$ functions. $~\displaystyle\sum_{n=-\infty}^\infty a^{n^2}=\theta_3(0,a)$. They are not known to possess a closed form, albeit they may be approximated with the help of the Gaussian integral. They play a very important part in number theory, and the great mathematical genius Ramanujan made great use of them, even inventing his own. Hope this helps.