Chapter 9.3, Problem 53E

Proof Let f be a positive, continuous, and decreasing function for $x\ge 1$, such that $a_n=f(n)$. Prove that if the series

${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{a}_n}$

converges to S, then the remainder $R_N=S-S_N$ is bounded by

$0\le R_N\le {\displaystyle {\int }_N^{\infty }f(x)dx}$