For which positive integers k is the following series convergent? (To enter - or , type -INFINITY or INFINITY.)
Summation of n=1 to infinity of (n!)^2 / (kn)!
ratio test: limit n-->infinity of [((n+1)!)^2/(kn+1)!] / [(n!)^2 / (kn)!] (have the original equation's n's replaced with n+1 and then divide that by the original equation)
The Attempt at a Solution
I am getting lost in how to simplify everything in order to find a convergence (or not). I have limit n --> infinity of [((n_1)!)^2 / (kn+1)!] * (kn)!/(n!)^2. Basically, I am multiplying by the reciprocal. I turned (kn+1)! into (kn+1)(kn!) and canceled out the other (kn)!, and I know I need to do something similar to (n!)^2, but I am not sure what to do?