- #1

Euge

Gold Member

MHB

POTW Director

- 2,073

- 242

If ##n## and ##k## are positive integers, let ##S_k(n)## be the sum of ##k##-th powers of the first ##n## natural numbers, i.e., $$S_k(n) = 1^k + 2^k + \cdots + n^k$$ Evaluate the limits $$\lim_{n\to \infty} \frac{S_k(n)}{n^k}$$ and $$\lim_{n\to\infty} \left(\frac{S_k(n)}{n^k} - \frac{n}{k+1}\right)$$

Last edited: