Question about polynomial $\sum_{j=1}^n j^k$

Question

How could I prove that $ 1^k + 2^k + \cdots + n^k \in \Theta(n^{k+1}) $ or, equivalently, $$ 0 < \lim_{n\to\infty}\frac{\sum_{i=1}^n i^k}{n^{k+1}} < \infty? $$ I would appreciate a hint rather than a solution. Thanks in advance. (I am sorry if this question is duplicate, I've searched but didn't found anything similar)

Answer 1

Think about Riemann sums (if you want the precise value of the limit, else Thomas' estimation does it!)

Answer 2

Hint: One way you can do this is to bound the sum on either side by an integral.

The function $f(x)=x^k$ is increasing on $[0,\infty)$; so, in particular, for each $j$ we have $$ \int_{j-1}^{j}x^k\,dx\leq j^k\leq\int_{j}^{j+1}x^k\,dx $$