If K is positive integer of the series, find the radius of the convergence

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- Jul 22nd 2007, 12:25 PMcamherokidSequences and series
If K is positive integer of the series, find the radius of the convergence

- Jul 22nd 2007, 01:44 PMThePerfectHacker
$\displaystyle \frac{(n!)^k}{(kn)!}$

Ratio test,

$\displaystyle \frac{[(n+1)!]^k}{(kn+k)!}\cdot \frac{(kn)!}{(n!)^k}$

Cancel a lot of stuff,

$\displaystyle \frac{(n+1)^k}{(kn+k)(kn+k-1)...(kn+1)}$

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

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

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

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