What is the radius of convergence for the series with $k$ as a positive integer?

[Chris L T521]

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]

This week's question was correctly answered by Sudharaka. You can find his solution below.

Spoiler

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, \(k^k\).