Chapter 10.5, Problem 67E

An early limit Working in the early 1600s, 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

$\underset{n\to \infty }{\lim }\frac{1}{n}{\displaystyle \underset{k=0}{\overset{n-1}{\sum }}{\left(\frac{k}{n}\right)}^p=\frac{1}{p+1}.}$

Use what you know about Riemann sums and integrals to verify this limit.