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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb690b5d7-6977-430e-9858-a5c74acd700c%2Ffe56b657-ad1b-4481-a702-c812a781f55f%2Fnb0mxik_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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