# Sequences and series

• Jul 22nd 2007, 01:25 PM
camherokid
Sequences and series
If K is positive integer of the series, find the radius of the convergence
• Jul 22nd 2007, 02:44 PM
ThePerfectHacker
$\frac{(n!)^k}{(kn)!}$
Ratio test,
$\frac{[(n+1)!]^k}{(kn+k)!}\cdot \frac{(kn)!}{(n!)^k}$
Cancel a lot of stuff,
$\frac{(n+1)^k}{(kn+k)(kn+k-1)...(kn+1)}$

Now the numerator is a polynomial of degree $k$ with leading coefficient $1$.

The denominator is a polynomial of degree $k$ and leading coefficient $k^k$.

So the limit of this ratio is $\frac{1}{k^k}$.

Which means the radius of convergence is $k^k$.