How to prove that: $\lim_{n\to\infty} \frac{1^n+2^n+\cdots+n^n}{n^n} = \frac{e}{e-1}$

Question

How to prove that:

$$\lim_{n\to\infty} \frac{1^n+2^n+\cdots+n^n}{n^n} = \frac{e}{e-1}$$

Any help is appreciated.