Proof Let be a positive, continuous, and decreasing function for such that Prove that if

Chapter 9, Problem 53

(choose chapter or problem)

Proof 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  

\(0 \leq R_{N} \leq \int_{N}^{\infty} f(x) d x\)

Text Transcription:

x geq 1

a_n=f(n)

sum_n=1^infinity a_n

R_N=S-S_N

0 leq R_n leq Int_N^infinity f(x) dx