Question

Find the radius of convergence of the power series, where c>0 and k is a positive integer

Answer 1

\[\sum_{0}^{\infty}\frac{ (n!)^k*x^n }{ (kn)! } \]

Answer 2

I apply the ratio test, and simplfy what I can and then I reach this point:

Answer 3

\[|x|\lim_{n \rightarrow \infty} \frac{ (n+1)^k * kn! }{ (k+kn)!}\]

Answer 4

Now I am stuck here

Answer 5

how do i simply this

Answer 6

\[\sum_{n=0}^\infty \frac{(n!)^kx^n}{(kn)!}\]
Ratio test:
\[\lim_{n\to\infty} \left|\frac{((n+1)!)^kx^{n+1}}{(k(n+1))!}\cdot\frac{(kn)!}{(n!)^kx^n}\right|\\
|x|\lim_{n\to\infty} \frac{(n+1)^k}{(kn+k)!}\cdot(kn)!\\
|x|\lim_{n\to\infty} \frac{(n+1)^k}{(kn+k)(kn+k-1)\cdots(kn+1)(kn)!}\cdot(kn)!\\
|x|\lim_{n\to\infty} \frac{(n+1)^k}{(kn+k)(kn+k-1)\cdots(kn+1)}\]
There are \(k\) factors in the denominator, so expanding the denominator would give you a polynomial starting with \(k^kn^k\). The highest power in the numerator is \(n^k\), so the limit would be
\[|x|\frac{1}{k^k}\]
By the ratio test, this series will converge if \(|x|\dfrac{1}{k^k}<1\), or \(|x|<k^k\).