Radius of convergence of power series which has factorial term

Question

I am trying to find radius of convergence of the following power series: $\sum_{n\geq 1} n^n z^{n!}$

I tried ratio test but it became complicated, I have never seen such radius of convergence problem with factorial.

Please help.

Answer 1

By Cauchy's criterion we have $R=1$. EDIT $\alpha_n=\sqrt[n]{n^n|z|^{n!}}=n|z|^{(n-1)!}$, if $|z|>1$ it is clear that $\alpha_n\to \infty$ (the series diverge). If $|z|<1$, then $\alpha_n=\left[\dfrac{n}{(n-1)!}\right]\left[(n-1)!|z|^{(n-1)!}\right]\to 0$ (the series converge).

Answer 2

By the Cauchy-Hadamard formula; the radius of convergence of a power series isgiven by $$\frac1{R}=\limsup |a_n|^{1/n}$$

Now , we may plug in and see that $|n^n|^{1/n!} \to 1 $ as $ n\to \infty $ .So we conclude that $R=1$.