Let f: [a, b] → R be a bounded function. Then f [a, b] → R is Riemann integrable if and only if there is a sequence {Pn} of partitions of [a, b] such that lim (U(f, Pn) - L(f, Pn)) = 0. nx In this case, the Riemann integral of f over [a, b] can be computed by S f = a lim L(f, Pn) = lim U(f, Pn). n4x n→∞ (4.1) (4.2)
Let f: [a, b] → R be a bounded function. Then f [a, b] → R is Riemann integrable if and only if there is a sequence {Pn} of partitions of [a, b] such that lim (U(f, Pn) - L(f, Pn)) = 0. nx In this case, the Riemann integral of f over [a, b] can be computed by S f = a lim L(f, Pn) = lim U(f, Pn). n4x n→∞ (4.1) (4.2)
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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please prove every step and don't use other one's answer thank yoi
![Let f [a, b] → R be a bounded function. Then f [a, b] → R is Riemann
!
integrable if and only if there is a sequence {Pn} of partitions of [a, b] such
that
lim (U(f, Pn) L(f, Pn)) = 0.
(4.1)
n→∞
In this case, the Riemann integral of f over [a, b] can be computed by
a
1
= lim L(f, Pn)
n-x
=
lim U(f, Pn).
n→∞
(4.2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a7d82e4-2bf5-4850-ba06-9621bb0bd1d8%2F593e37a4-a9aa-4a1d-a60f-0b722658bd4d%2Fbc874tj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let f [a, b] → R be a bounded function. Then f [a, b] → R is Riemann
!
integrable if and only if there is a sequence {Pn} of partitions of [a, b] such
that
lim (U(f, Pn) L(f, Pn)) = 0.
(4.1)
n→∞
In this case, the Riemann integral of f over [a, b] can be computed by
a
1
= lim L(f, Pn)
n-x
=
lim U(f, Pn).
n→∞
(4.2)
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