Limit of $\frac{1^n+2^n+\cdots+(n-1)^n}{n^n}$ as $n$ go into infinity

Question

$$\lim_{n\to\infty}\frac{1^n+2^n+\cdots+(n-1)^n}{n^n}$$

I have found that the sequence is increasing as $n$ increases. I guess that the answer should be $\frac{1}{1-e}$.