Exponential limits

Relevant wiki: Riemann Sums

$\begin{aligned} L & = \lim_{n \to \infty} \frac {1^p+2^p+3^p + \cdots + n^p}{n^{p+1}} \\ & = \lim_{n \to \infty} \sum_{k=1}^n \frac {k^p}{n^{p+1}} \\ & = \lim_{n \to \infty} \frac 1n \sum_{k=1}^n \left(\frac kn\right)^p & \small \color{#3D99F6} \text{By Riemann sums} \\ & = \int_0^1 x^p dx \\ & = \frac {x^{p+1}}{p+1} \ \bigg|_0^1 \\ & = \boxed{\dfrac 1{p+1}} \end{aligned}$