$\sum_{n=1}^\infty z^{n!}$
Here is what I've got so far
Claim: The above series converges for $|z|<1$.
Pick $|z|
So $\sum\limits_{n=1}^\infty r^{n!}$ is a majorant for $\sum\limits_{n=1}^\infty z^{n!}$.
$\sum\limits_{n=1}^\infty r^{n!}$ is a real series so we can test for convergence.
This is where I get stuck, I've tried the ratio test but that doesn't seem to work and I can't think of a function that would work for the comparison test.