# Problem of the Week #63 - June 10th, 2013

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#### Chris L T521

##### Well-known member
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Thanks again to those who participated in last week's POTW! Here's this week's problem!

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Problem: If $k$ is any positive integer, determine the radius of convergence for the series $\displaystyle\sum_{n=0}^{\infty}\frac{(n!)^k}{(kn)!}x^n$.

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#### Chris L T521

##### Well-known member
Staff member
This week's question was correctly answered by Sudharaka. You can find his solution below.

Let, $$\displaystyle a_n=\frac{(n!)^k}{(kn )!}x^n$$. By the Ratio test, the series converges if,

$\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|<1$

$\Rightarrow |x|<\lim_{n\rightarrow\infty}\left|\frac{[k(n+1)]\times [k(n+1)-1]\times \cdots\times [kn+1]}{(n+1)^k}\right|=k^k$

$\therefore |x|<k^k$

Hence the radius of convergence is, $$k^k$$.

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