Show that \begin{equation*} \sum_{n=0}^{+\infty}x^{n^{2}} \end{equation*} is equivalent to \begin{equation*} \frac{1}{2}\sqrt{\frac{\pi}{1-x}} \end{equation*} as $x\in (0,1)$ approaches $1$.
This comes from a French complex analysis book, and normally the answer should not use any fancy theorems, since so far the authors have only covered Cauchy's integral formula.