5.1.3. Let f: [a, b] → R be a bounded function. Suppose there exists a sequence of partitions {Pr} of [a, b] such that lim (U(PR. f) – L(Pr, f)) = 0 Show that f is Riemann integrable and that f = lim U(Pk, f) = lim L(Pk, f).
5.1.3. Let f: [a, b] → R be a bounded function. Suppose there exists a sequence of partitions {Pr} of [a, b] such that lim (U(PR. f) – L(Pr, f)) = 0 Show that f is Riemann integrable and that f = lim U(Pk, f) = lim L(Pk, f).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![5.1.3. Let f: [a, b] → R be a bounded function. Suppose there exists a sequence of partitions {P} of
[a, b] such that
lim (U(PR f) - L(Px, f)) = 0
k→∞
Show that f is Riemann integrable and that
b
f = lim U(Pr,f) = lim L(P, f).
k»
a
Prove that f E R[-1,1] and compute f, f using the definition of the integral (But feel free to use the
propositions of this section).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0548f28d-2867-4ce8-91e2-809a65372be3%2Fa9f9c4ae-da1d-48a1-bb46-2e541eba10ad%2Fn4qdqd_processed.png&w=3840&q=75)
Transcribed Image Text:5.1.3. Let f: [a, b] → R be a bounded function. Suppose there exists a sequence of partitions {P} of
[a, b] such that
lim (U(PR f) - L(Px, f)) = 0
k→∞
Show that f is Riemann integrable and that
b
f = lim U(Pr,f) = lim L(P, f).
k»
a
Prove that f E R[-1,1] and compute f, f using the definition of the integral (But feel free to use the
propositions of this section).
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