$$\sum_{n=1}^\infty z^{n!}$$
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.