Hello
Problem:
For which positive integer k is the following series convergent?
$\displaystyle \sum_{n=1}^{\infty} \frac{(n!)^2}{(kn)!}$
The Ratio Test seems a good choice.
After applying the Ratio Test, I will face the following limit:
$\displaystyle \lim_{n\to\infty} \frac{(n+1)^2(kn)!}{(kn+k)!}$
Then ?