Limit $\lim_{n \to \infty} \frac{n!}{n^n}$

Question

How can I compute this limit: $\lim_{n \to \infty} \frac{n!}{n^n}$. I have tried taking the n-th root but then I have problem managing the factorial. Thanks to everyone who will help!

Answer 1

HINT

By ratio test

$$\frac{(n+1)!}{(n+1)^{n+1}}\frac{n^n}{n!} =\frac{1}{\left(1+\frac1n\right)^n}$$

and recall that $\left(1+\frac1n\right)^n\to e$.

Answer 2

Let $a_n=\frac{n!}{n^n}$

Prove that $\frac{a_{n+1}}{a_n} \to \frac{1}{e}<1$

Thus from ratio test for sequences we have that $a_n \to 0$

Answer 3

You don't have to involve a root or ratio test here. $$ \frac{n!}{n^n} = \frac{1 \times 2 \times \cdots \times n}{n \times n \times \cdots \times n} = \frac1n \times \frac2n \times \cdots \times \frac nn < \frac 1n. $$