Theta function: Absolute convergence

Question

I have to proof that the series $$ \theta (z)= \sum_{n=-\infty}^{+\infty} e^{\pi i [n^{2} \tau + 2nz]}$$ converges absolutely and uniformly on compact subsets of $\mathbb{C}$. Necessary and sufficient condition for the absolute and uniform convergence of the series is that for all values ​​of z, the terms of the series are all limited in domain A. Is there any series $\sum b_n(z)$ such that $\theta(z) \le \sum b_n (z)$?