Ramanujan defined a now famous q-series as
$$\sum_{n=-\infty}^{\infty}q^{n^2} = \left(-q;q^2\right)^2_{\infty}\left(q^2;q^2\right)_{\infty}$$
I wanted to prove this identity but I wasn't sure where to begin. I tried looking for an identity involving $$\sum_{n=-\infty}^{\infty}q^{k}$$ for $k=n^2$ however it doesn't seem to converge and so I was stuck at this. I also attempted to work backwards starting with a decomposition of $\left(-q;q^2\right)^2_{\infty}\left(q^2;q^2\right)_{\infty}$ but that also quickly led nowhere. Does anyone know how to prove this?