radius of convergence of power series n!z^n/n^n

Question

Please help me find radius of convergence of the following: check the attached picture to have a better image \sum n=0 to infinity n! z^n / n^n I used Hadamard's formula 1/R=lim sup|an|^1/n didnt know how to proceed is the radius = infinity? or 1/e plz help

Answer 1

Hint: Use the root test with $a_{n}=(n!/n^{n}) z^{n}$ to get $$ \limsup\sqrt[n]{|a_{n}|}=z\lim\frac{\sqrt[n]{n!}}{n} $$