Limit of sequence $n!\left(\frac{e}{n}\right)^n$

Question

Find the limit of $$ \lim_{n\to +\infty} n!\left(\frac{e}{n}\right)^n. $$ I have shown that $u_{n+1}>u_n$, but I am not sure where to go from here.

Answer 1

By Stirling's approximation we have:

$$n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$$

So we can make comparisons to get:

$$\lim_{n\to +\infty} n!\left(\frac{e}{n}\right)^n \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \times \left(\frac{e}{n}\right)^n$$

Merge the two $\left(\frac{n}{e}\right)^n \times \left(\frac{e}{n}\right)^n \rightarrow (1)^n$. Can you figure out what to do now?

Answer 2

Let $a_n=n!(e/n)^n$. You can see $a_{n+1}/a_{n}>e$, so $lim_{n \rightarrow \infty}a_n$ doesn't exit.