Solved: Let f be a positive, continuous, and decreasing function for such that Prove

Chapter 9, Problem 67

(choose chapter or problem)

Let f be a positive, continuous, and decreasing function for \(x \geq 1\) such that \(a_{n}=f(n)\). Prove that if the series

\(\sum_{n=1}^{\infty} a_{n}\)

converges to S, then the remainder \(R_{N}=S-S_{N}\) is bounded by

\(0\ \le\ R_N\ \le\ \int_N^{\infty}\ f(x)\ dx\)

Text Transcription:

x geq 1

a_n = f(n)

sum_n=1 ^infty a_n

R_N = S - S_N

0 leq R_N leq int_N ^infty f(x) dx