How can I find $\lim_{n\to \infty}\left(\cfrac{1^p+2^p+\cdots+ n^p}{n^p}-\cfrac{n}{p+1}\right)$?

Question

Please help me to find the following limit:

$$\lim_{n\to \infty}\left(\cfrac{1^p+2^p+\cdots +n^p}{n^p}-\cfrac{n}{p+1}\right),$$ where $p\in \mathbb{N}$.

Answer 1

By the trapezoidal method $$ \frac1n\left(\tfrac12f(0)+\sum_{k=1}^{n-1}f(\tfrac{k}{n})+\tfrac12f(1)\right)=\int_0^1 f(x)\,dx+O(\tfrac1{n^2}). $$ For $f(x)=x^p$ we get $$ \frac1{n^{p+1}}\left(\sum_{k=1}^n k^p - \tfrac12 n^p\right)=\frac{1}{p+1}+O(n^{-2}) $$ and in the form of the task $$ \frac{1^p+2^p+…+n^p}{n^p}-\frac{n}{p+1}=\frac12 + O(n^{-1}) $$