by Consider the sequence of functions (fn) defined on the interval [0, 1] given n²x², 0≤x≤ 1/n 1, 1/n < x ≤ 1 În(x) = { a) Sketch the graphs of fn for n = 1, n = 2 and n = 3. b) Let g(x) = 1 for 0 ≤ x ≤ 1. Show that (fn) converges to g in the mean. c) Let 0, for x = 0 h(x) = { 1, 0 < x < 1 Prove that (fn) converges pointwise to h.
by Consider the sequence of functions (fn) defined on the interval [0, 1] given n²x², 0≤x≤ 1/n 1, 1/n < x ≤ 1 În(x) = { a) Sketch the graphs of fn for n = 1, n = 2 and n = 3. b) Let g(x) = 1 for 0 ≤ x ≤ 1. Show that (fn) converges to g in the mean. c) Let 0, for x = 0 h(x) = { 1, 0 < x < 1 Prove that (fn) converges pointwise to h.
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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Consider the sequence of functions (fn) defined on the interval [0, 1] given
Sn(x) = {
fn
a) Sketch the graphs of fn for n = 1, n = 2 and n = 3.
b) Let g(x) = 1 for 0 ≤ x ≤ 1. Show that (fn) converges to g in the mean.
c) Let
h(x)
n²x², 0 ≤ x ≤ 1/n
1, 1/n < x ≤ 1
=
={
0, for x = 0
1, 0 < x < 1
Prove that (fn) converges pointwise to h.
d) Does the sequence (fn) converge uniformly to h? No credit will be given for any
unjustified answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6a5ac2db-d584-49da-a0ed-efa785bd6c28%2Fb9258cd9-ca3f-4af9-850a-f01821ddf3c3%2Fyx69h44_processed.jpeg&w=3840&q=75)
Transcribed Image Text:by
Consider the sequence of functions (fn) defined on the interval [0, 1] given
Sn(x) = {
fn
a) Sketch the graphs of fn for n = 1, n = 2 and n = 3.
b) Let g(x) = 1 for 0 ≤ x ≤ 1. Show that (fn) converges to g in the mean.
c) Let
h(x)
n²x², 0 ≤ x ≤ 1/n
1, 1/n < x ≤ 1
=
={
0, for x = 0
1, 0 < x < 1
Prove that (fn) converges pointwise to h.
d) Does the sequence (fn) converge uniformly to h? No credit will be given for any
unjustified answer.
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