# Finding the radius of convergence of $\sum_{0}^{\infty}z^{n!}$ ?

I was thinking about the problem that says: What is the radius of convergence of $\sum_{0}^{\infty}z^{n!}$ ?

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22 minutes.  – Did Dec 7 '12 at 6:13

You mean the radius of convergence of $$\sum_{0}^{\infty}z^{n!}$$
We have $a_{n!}=1$ for all $n!$. Then by Hadamard's theorem we have $$\lim_{n=m!,n\rightarrow \infty}\sup (1)^{1/n}=1$$
Hadamard is overkill, no? Isn't it more-or-less trivial that it diverges at $z=1$, and converges (by comparison to $\sum z^n$) for $|z|\lt1$? – Gerry Myerson Dec 7 '12 at 4:56