Thread: Find value of k for which the series converges.

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 ?

(kn+k)! = (kn+k)...................(kn+1)(kn)!


you have (n+1)^2/[(kn+k)...................(kn+1)

for k = 2

lim(n+1)^2/ [2(n+1)(2n+1)] = 1/4 and we have convergence

For k > 2 limit is 0 as denominator is of order 3 or higher

for k= 1 lim(n+1)^2/(n+1) = infinity