Solution Found!
An early limit Working in the early 1600s, the
Chapter 12, Problem 78AE(choose chapter or problem)
An early limit Working in the early 1600s, the mathematicians Wallis, Pascal, and Fermat were attempting to determine 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.
Questions & Answers
QUESTION:
An early limit Working in the early 1600s, the mathematicians Wallis, Pascal, and Fermat were attempting to determine 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.
ANSWER:Problem 78AEAn early limit Working in the early 1600s, the mathematicians Wallis. Pascal, and Fermat were attempting to determine the area of the region under the curve y = xp between x = 0 and x = 1, where p is a positive integer