For which positive integers k is the following series convergent?
[(n!)^2]/[(kn)!] as n goes from 1 to infinity. Please show the steps using the ratio test.
$\displaystyle \dfrac{{\left[ {\left( {n + 1} \right)!} \right]^2 }}
{{\left( {kn + k} \right)!}}\dfrac{{\left( {kn} \right)!}}
{{\left( {n!} \right)^2 }} = \dfrac{{\left( {n + 1} \right)^2 \left( {n!} \right)^2 }}
{{\left( {n!} \right)^2 }}\dfrac{{\left( {kn} \right)!}}
{{\left( {kn + k} \right)!}}$
If $\displaystyle k=1$ we get $\displaystyle \dfrac{1}{n+1}$
If $\displaystyle k=2$ we get $\displaystyle \dfrac{(2n)!}{(2n+2)!}=\dfrac{1}{(2n+2)(2n+1)}$
If $\displaystyle k=3$ we get $\displaystyle \dfrac{(3n)!}{(3n+3)!}=\dfrac{1}{(3n+3)(3n+2)(3n+1 }$
Get it?