Radius of convergence of this power series We're given the power series $$ \sum_1^\inf \frac{j!}{j^j}z^j$$
and are asked to find radius of convergence R. I know the formula $R=1/\limsup(a_n ^{1/n})$, which leads me to compute $\lim \frac{j!^{1/j}}{j}$, and then I'm stuck.
The solution manual calculates R by $1/\lim|\frac{a_{j+1}}{a_j}|$, but I can't figure out the motivation for that formula.

Use Stirling's formula
$$n! \sim {n^n e^{-n}\over\sqrt{2\pi n}}.$$
