What's $\sum_{n \ge 0} q^{n^2}$? Is there a relatively simple way of calculating the sum of the series $$\sum_{n=0}^\infty q^{n^2}, \quad |q|<1 ?$$

This is can be written in terms of the Jacobi theta function, $$\vartheta_{3}(q)=\sum_{n=-\infty}^{\infty}q^{n^{2}}.$$  Specifically, your function is $\frac{1}{2}\left(\vartheta_{3}(q)+1\right).$  I am not sure what you mean by calculating the series for $q<1$, but there are several known special values, for example $\theta_3\left(e^{-\pi}\right)=\pi^{1/4}\Gamma\left(3/4\right)^{-1},$ and it satisfies the functional equation $$\vartheta_{3}\left(e^{-\pi x}\right)\vartheta_3\left(e^{-\pi/x}\right)^{-1}=\frac{1}{\sqrt{x}}. $$
See also http://en.wikipedia.org/wiki/Theta_function#Jacobi_theta_function for more information.
