I would like to know if I didn't make a mistake in determining the radius of convergence for the following series: (a) $\sum_{n=0}^\infty \frac{z^{n^2}}{n!}$ and (b) $\sum_{n=0}^\infty \frac{z^{n^2}}{(n!)^n}$.
For (a) the ratio test yields
$$ \lim_{n\rightarrow \infty}\left|\frac{z^{(n+1)^2} n!}{(n+1)! z^{n^2}} \right|= \lim_{n\rightarrow\infty}\frac{|z|^{2n+1}}{n+1}=0 $$
if and only if $|z|\leq1$. Thus the radius of convergence is one.
For (b) using the root test I have
$$ \lim_{n\rightarrow\infty}\frac{|z|^n}{n!}=0 $$
if and only if again $|z|\leq1$ and the radius of convergence is again one.
For b) the limit as $n$ approaches infinity for the formulas is $0$ regardless of what $|z|$ is since $n!$ is in the denominator and overtakes the exponential in the numerator its growth.
I think your solution for a) is correct though since the exponential term in the top will dominate so you need that to go to $0$.