Consider the series $$\sum_{n=-\infty}^{\infty} e^{\pi i [n^2 \tau + 2n z]},$$ where Im$(\tau) >0$ and $z \in \mathbb{C}$. Prove that this series converges absolutely and uniformly on compact subsets of $\mathbb{C}$.
The compact subsets of $\mathbb{C}$ are those subsets which are closed and bounded. Moreover, with some algebraic manipulation, we obtain the following:
$$\left| \sum_{n=-\infty}^{\infty} e^{\pi i [n^2 \tau + 2nz]} \right| \leq \left| e^{\pi i n^2 \tau} \right| \cdot \left| \sum_{n=-\infty}^{\infty} e^{2 \pi i nz} \right| \leq \sum_{n=-\infty}^{\infty} \left| e^{2\pi i} \right|^{nz} \leq \sum_{n=-\infty}^{\infty} 1^{nz}.$$ From here however, I'm unsure of how to continue.