For which positive integer k is the following series convergent?

Question

(n!)^2/(kn)!

i know you use the ratio test but where do you go from there?

Answer 1

Since you limit the discussion to positive integers, the problem becomes more straightforward.

Let k = 1, the series's term can be written as n! and clearly, the series diverges.

Let k = 2, the series's term can be written as

(n!)^2/(2n*(2n-1)*...(n+1)
*n*(n-1)...*1)
=n!/(2n*(2n-1)*...*(n+1))

The above tends to zero with n tending to infinity. So the series converges.

Clearly, for k>2, the series also converges (because each term is smaller than the corresponding term when k=2 and is always positive.)

So the answer: k = 2, 3, 4, ...

Answer 2

ratio test
a_n+1/a_n = (n+1)^2/(kn+1)*(kn+2)****(kn+k)
the limit is =0 for every k>2 and 1/k^2 for k=2
so for all integers >=2 the series is convergent