I was looking at the power series $\sum\frac{z^n}{n!}$ and $\sum n!z^n$, and wanted to compute their radii of convergence.
For the first, $\limsup \sqrt[n]{1/n!})=0$, and for the second $\limsup \sqrt[n]{n!}=\infty$, so the radii are $\infty$ and $0$ respectively.
I'm curious, without resorting to Wolfram or Mathematica or some similar program, how could one justify by hand these limits? Intuitively they make sense that they are what they are, since the factorial should grow more quickly than the power of $1/n$ can handle. I suppose would explain the other since the terms are just reciprocals. Thanks!