1. Let fn (0, ∞) → R where fn(x) (0, ∞)→ R where f(x) = 0. = nx for all n E N and let f (a) Graph f1, f2, f3, and f on the same axes. (b) Prove that fn converges pointwise to f on (0, ∞). (c) Does fn converge uniformly to f on (0, ∞o)? Prove your answer.
1. Let fn (0, ∞) → R where fn(x) (0, ∞)→ R where f(x) = 0. = nx for all n E N and let f (a) Graph f1, f2, f3, and f on the same axes. (b) Prove that fn converges pointwise to f on (0, ∞). (c) Does fn converge uniformly to f on (0, ∞o)? Prove your answer.
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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Question
Transcribed Image Text:1. Let fn (0, ∞) → R where fn(x)
(0, ∞) → R where f(x) = 0.
=
nx
for all ne N and let f
(a) Graph f1, f2, f3, and f on the same axes.
(b) Prove that fn converges pointwise to f on (0, ∞).
(c) Does fn converge uniformly to f on (0, ∞o)? Prove your answer.
(d) Does fn converge uniformly to f on [1, ∞)? Prove your answer.
![(0,0)
Ex.
Let S = [0, 1] x fn(x) = x^ *nexo & 0≤x²1
or 5 to f(x) = { 1 + x = 1
Jaim: {fr} Conv. ptwise
(₁,1)
f(x)=x² = x to ¼/a ft = (1/2)
fa (x) = x²
fa= 1/4
f3 (x)=x²
9 f¶ (x) = .
=X
lim
So txES, 200
limm
Eanx"
lim
Bf Suppose 0≤x²1 - Then apo fn(x) = (im X"=0
Cim
lim
1^ ==
Suppose X=1. Then
2-01-1
4
sao fn(x) =
fn (x) exists,
S₁ = 90 x°
S₂ = 96x° +9₁x = 90 +9₁ x
S3 = ax +9₁ x + 9₂x²
= 90
+3 = ½
Does {fr} Conv. pluise on s? nf
Show of = 1
0
Sofn (x) dx = Area of A = 1/2 ² 1/0 +0=1
but So f(x) dx=0 lim
S = [0, 1] for 122, define fn (x) = ['n³x if 0≤x≤/n
2-n²(x - ²) 2 % ≤ x ≤
-0 i 3/ / LX ≤I
fn
15-01-
din f
Lim So fm (x) dx # Sr f(x) dx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2Fb1850726-c7e3-4a6d-ad7d-33981e06e13b%2Fxd10o4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(0,0)
Ex.
Let S = [0, 1] x fn(x) = x^ *nexo & 0≤x²1
or 5 to f(x) = { 1 + x = 1
Jaim: {fr} Conv. ptwise
(₁,1)
f(x)=x² = x to ¼/a ft = (1/2)
fa (x) = x²
fa= 1/4
f3 (x)=x²
9 f¶ (x) = .
=X
lim
So txES, 200
limm
Eanx"
lim
Bf Suppose 0≤x²1 - Then apo fn(x) = (im X"=0
Cim
lim
1^ ==
Suppose X=1. Then
2-01-1
4
sao fn(x) =
fn (x) exists,
S₁ = 90 x°
S₂ = 96x° +9₁x = 90 +9₁ x
S3 = ax +9₁ x + 9₂x²
= 90
+3 = ½
Does {fr} Conv. pluise on s? nf
Show of = 1
0
Sofn (x) dx = Area of A = 1/2 ² 1/0 +0=1
but So f(x) dx=0 lim
S = [0, 1] for 122, define fn (x) = ['n³x if 0≤x≤/n
2-n²(x - ²) 2 % ≤ x ≤
-0 i 3/ / LX ≤I
fn
15-01-
din f
Lim So fm (x) dx # Sr f(x) dx
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