Calculating $\lim_{n \to \infty}\left(\frac{1^n +2^n +3^n + \cdots + n^n}{n^n}\right)$ [duplicate]

Evaluate :

$$\lim_{n \to \infty}\left(\frac{1^n +2^n +3^n + \cdots + n^n}{n^n}\right)$$

I tried using squeeze theorem, but I couldn't find the proper inequality. I also thought of using the Stolz–Cesàro Theorem but it further complicated the problem.

Any help will be appreciated.
Thanks.

marked as duplicate by Daniel FischerMay 10 '16 at 15:01

• $$\sum_{k = 0}^{n-1} \biggl( 1 - \frac{k}{n}\biggr)^n.$$ I think it has been asked before. Let me look. – Daniel Fischer May 10 '16 at 14:59