For which positive integers k is the following series convergent? (Enter your answer as an inequality.)

Question

∞	(n!)2
(kn)!
n = 1

Answer 1

It's not clear if you mean the summand to be (n!)²/(kn)! or its reciprocal.

For k = 1, (n!)²/(kn)! → ∞ as n → ∞ so that sum would diverge.

For k > 1, by the Ratio Test

r = (((n+1)!)²/(kn+k)!)/((n!)²/(kn)!)

= (n + 1)²/((kn+k)(kn+k–1)...(kn+1))

= (n + 1)²/n^k((k+k/n)(k+(k–1)/n)...(k+1/n)).

For k > 2, clearly r → 0 as n → ∞, so the series converges.

For k = 2, r → 1/(2×2) = 1/4 < 1 as n → ∞, so the series converges.