Radius of convergence of $\sum_{n \geq 0}a_n z^{n!}$ given that of $\sum_{n \geq 0}a_n z^{n}$

Question

What would be the radius of convergence of $\sum_{n \geq 0}a_n z^{n!}$ given that the radius of convergence of $\sum_{n \geq 0}a_n z^{n}$ is $L$.

Answer 1

Hint: let $b_n=a_k$ if $n=k!$ and $0$ if $n$ is not a factorial. We have to find the radius of convergence of $\sum b_nz^{n}$. Use root test. Can you see that the radius of convergence is $1$ provided $0<L<\infty$?.

Answer 2

Using the Cauchy-Hadamard theorem:

$r=\frac1{\limsup_{n\to\infty}\sqrt[n!]{a_n}}=\frac1{\limsup_{n\to\infty}\sqrt[(n-1)!]L}=1$, for $L\neq0,\infty$.