Q(A). Let {fn(x)}1 =be a sequence of functions1+ (x – 2)"defined over [2,3]. Show that:(a) fn(x) is meaurable and monotonic increasing for all n.(b) {fn(x)}-1 converges a.e. to a function f(x) to be determined.(c) Apply the monotone convergence theorem to evaluate Lim fn(x)dµ.(B) Let {fn(x)}1 = {x ln x. cos(x – 1)"}1be a sequence of meaurable functions defined over [1,2]. Show that:(a) {fn(x)}1 converges to a function f(x) to be determined.(b) Show that fn(x) is dominated by some integrable function for all n.Then apply the dominated convergence theoremto evaluate Limx In x.cos(x- 1)"dµ.

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Description: Q(A). Let {fn(x)}1 =be a sequence of functions1+ (x – 2)"defined over [2,3]. Show that:(a) fn(x) is meaurable and monotonic increasing for all n.(b) {fn(x)}-1 converges a.e. to a function f(x) to be determined.(c) Apply the monotone convergence theo

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Q(A). Let {fn(x)} = be a sequence of functions 1+ (x – 2)" defined over [2,3]. Show that: (a) fn(x) is meaurable and monotonic increasing for all n. (b) {Sn(x)}1 converges a.e. to a function f(x) to be determined. (c) Apply the monotone convergence theorem to evaluate Lim rtp(x)"S

Q(A). Let {fn(x)} = be a sequence of functions 1+ (x – 2)" defined over [2,3]. Show that: (a) fn(x) is meaurable and monotonic increasing for all n. (b) {Sn(x)}1 converges a.e. to a function f(x) to be determined. (c) Apply the monotone convergence theorem to evaluate Lim rtp(x)"S

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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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

Q(A). Let {fn(x)}1 =
be a sequence of functions
1+ (x – 2)"
defined over [2,3]. Show that:
(a) fn(x) is meaurable and monotonic increasing for all n.
(b) {fn(x)}-1 converges a.e. to a function f(x) to be determined.
(c) Apply the monotone convergence theorem to evaluate Lim fn(x)dµ.
(B) Let {fn(x)}1 = {x ln x. cos(x – 1)"}1
be a sequence of meaurable functions defined over [1,2]. Show that:
(a) {fn(x)}1 converges to a function f(x) to be determined.
(b) Show that fn(x) is dominated by some integrable function for all n.
Then apply the dominated convergence theorem
to evaluate Lim
x In x.cos(x- 1)"dµ.

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