TITLE: Limit. $\lim_{n \to \infty}\frac{1^p+2^p+\ldots+n^{p}}{n^{p+1}}$.
QUESTION [7 upvotes]: Have you any idea how to find the limit of the following sum: 
$$\lim_{n \to \infty}\frac{1^p+2^p+\ldots+n^{p}}{n^{p+1}}.$$
Stolz-Cesaro? any more ideas?

REPLY [8 votes]: Prove by induction on $p$ that $$f_p(n)=\sum_{k=0}^n k^p$$
is a polynomial of degree $p+1$ with leading term $$\frac{n^{p+1}}{p+1}$$
Immediately conclude the limit is $$\frac{1}{p+1}$$

Alternatively, use Stolz Cesaro, to show your limit is the same as $$\lim\limits_{n\to\infty}\frac{(n+1)^{p}}{(n+1)^{p+1}-n^{p+1}}$$
Then use that $$\lim_{x\to 0}\frac{(1+x)^{\alpha}-1}x=\alpha$$