What is the radius of convergence of $\sum_{n=0}^\infty z^{n!}$? Find the radius of convergence:
$$\sum_{n=0}^\infty z^{n!}$$
I suppose that this serie behaves in a similar way that $\sum_{n=0}^\infty z^{n}$ , so I have that it converges when $|z| < 1 $.
Could someone explain me if this approach is correct and give me a more rigorous proof?
Thanks in advance

If $|z|<1$, then $|z|^{n!} \leq |z|^n$ so the series converges. If $z=1$, then the series obviously diverges.
So the radius is $1$.
