An early limit Working in the early 1600 s, the mathematicians Wallis, Pascal, and Fermat were calculating the area of the region under the curve $y=x^{p}$ between $x=0$ and $x=1,$ where $p$ is a positive integer. Using arguments that predated the Fundamental Theorem of Calculus, they were able to prove that
$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n-1}\left(\frac{k}{n}\right)^{p}=\frac{1}{p+1}$$
Use what you know about Riemann sums and integrals to verify this limit.