What is the radius of convergence of the power series $\sum_{n=0}^\infty \frac {(n!)^k}{(kn)!}z^n$?

Question

What is the radius of convergence of the power series? $$\sum_{n=0}^\infty \frac {(n!)^k}{(kn)!}z^n$$

Progress

Used the ratio test, but got $0$ from it.

Answer 1

$$ \lim_{n\rightarrow \infty}\frac{c_n}{c_{n+1}} = \lim_{n\rightarrow \infty}\frac{(kn+1)(kn+2)\cdots(kn+k)}{(n+1)^k} = \lim_{n\rightarrow \infty}k^k\frac{(n+\frac1k)}{n+1}\cdots\frac{(n+1)}{n+1} = k^k. $$