$\limsup_{n\to \infty}(n^n/n!)^{1/n} = e$

Question

I'm trying to understand how $$\limsup_{n\to \infty}(n^n/n!)^{1/n} = e$$

i.e., the fact that the power series $$\sum_{n=1}^\infty (n^n/n!)z^n$$ has a radius of convergence of $\frac1e$

I haven't the slightest clue how to prove this, and am looking for any sort of nudge in the right direction, thanks.

Answer 1

Assuming you can't use Stirling's approximation (otherwise it's a one line exercise), have you tried using the ratio test? It should lead you towards a well known limit...