TITLE: Limit of $\frac{1}{\sqrt[n]{n!}}$ as $n$ approaches infinity
QUESTION [6 upvotes]: So i was trying to evalue this limit:
$$\lim_{n \to \infty}\frac{1}{\sqrt[n]{n!}}, n \in \mathbb{N}$$
This, of course, by common sense is equal to zero (since factorial grows a lot faster). Is there a way to prove this limit without having to tackle with proving function growth rate. I;m not sure how that would be done, but I believe i would have to expand the factorial function to set $\mathbb{R}$ in order to compare, and that's still beyond my abilities. Thanks.

REPLY [15 votes]: use the series $e^x = 1 + \dfrac{x}{1!} + \cdots + \frac{x^n}{n!} + \cdots$ to get  $$e^n > \dfrac{n^n}{n!}  $$ which gives you $$\left(\frac{1}{n!}\right)^{1/n} < \frac{e}{n} $$ and letting $n \to \infty$ gives you the limit zero.