\lim_{n\rightarrow+\infty}\sum_{k=1}^n\frac{k^p}{n^{p+1}}=\lim_{n\rightarrow+\infty}\sum_{k=1}^n\frac 1n\left(\frac kn\right)^p=\int_0^1x^p\,dx=\left\{
\begin{array}{ll}
\frac 1{p+1}, & p>-1,
\vspace{0.5em}\\
+\infty & p\leq-1.
\end{array}
\right.