If K is positive integer of the series, find the radius of the convergence
$\displaystyle \frac{(n!)^k}{(kn)!}$
Ratio test,
$\displaystyle \frac{[(n+1)!]^k}{(kn+k)!}\cdot \frac{(kn)!}{(n!)^k}$
Cancel a lot of stuff,
$\displaystyle \frac{(n+1)^k}{(kn+k)(kn+k-1)...(kn+1)}$
Now the numerator is a polynomial of degree $\displaystyle k$ with leading coefficient $\displaystyle 1$.
The denominator is a polynomial of degree $\displaystyle k$ and leading coefficient $\displaystyle k^k$.
So the limit of this ratio is $\displaystyle \frac{1}{k^k}$.
Which means the radius of convergence is $\displaystyle k^k$.