How to calculate radius of convergence of the following series? How can I calculate radius of convergence of the following series?
$$\Large
\sum\limits_{n=0}^\infty \frac{5^{n+1}}{\sqrt[n]{(2n)!}}z^{n}
$$
I tried using D'alembert convergence test but cannot figure out how to calculate.
I know the answer is $\LARGE\frac{1}{5}$

By Stirling's formula,
$$(2n!)^{1/n^2} \sim \left({(2n)^{2n}\over e^{2n}\sqrt{4\pi n}}\right)^{1/n^2
}\sim {(2n)^{2/n}\over e^{2/n} } \rightarrow 1 $$
as $n\to\infty$.  So the radius of convergence is 1/5.
