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
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
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![**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
\[
\lim_{n \to \infty} S_n = \int_a^b f(x) \, dx.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff906d8fa-9b6a-4046-ba2d-f959fe6282ae%2Fa3e243f9-6a29-4b83-a77f-2e6e3c542e8e%2F2wprl2s_processed.jpeg&w=3840&q=75)
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
\[
\lim_{n \to \infty} S_n = \int_a^b f(x) \, dx.
\]
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