2. Suppose that f is an integrable function on [a, b]. Suppose that for each n, Sn is a Riemann sum for f corresponding to a partition of width < 1/n. Prove that lim Sn | f(x) d.r. %3D n00

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
**Problem 2:**

Suppose that \( f \) is an integrable function on \([a, b]\). Suppose that for each \( n \), \( S_n \) is a Riemann sum for \( f \) corresponding to a partition of width less than \( 1/n \). Prove that

\[
\lim_{n \to \infty} S_n = \int_a^b f(x) \, dx.
\]
Transcribed Image Text:**Problem 2:** Suppose that \( f \) is an integrable function on \([a, b]\). Suppose that for each \( n \), \( S_n \) is a Riemann sum for \( f \) corresponding to a partition of width less than \( 1/n \). Prove that \[ \lim_{n \to \infty} S_n = \int_a^b f(x) \, dx. \]
Expert Solution
steps

Step by step

Solved in 2 steps

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,