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.

The ratio test gives that, if $\{a_n\}$ is a positive sequence and $\frac{a_{n+1}}{a_n}\to L$, then
$$ \lim_{n\to +\infty}\sqrt[n]{a_n} = L, $$
too. You may also notice that by Lagrange's inversion your series is related with the inverse function of $x\mapsto x\cdot e^x$, i.e. the Lambert $W$-function.
