Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent.
I use a root test:
$\lim_{n\rightarrow\infty} |\frac{e^{nz^2}}{n}|^{1/n}=\lim_{n\rightarrow\infty} |\frac{e^{nz^2}}{n}|^{1/n}=|e^{z^2}|$
And now I have to find all $z$ such that $|e^{z^2}|<1$.
Is that correct and how can I do that?