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.


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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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