(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