Check if the following sequences are uniformly convergent 2nx fn (2) = 1+n*_z, 2 € [0, 1]
Check if the following sequences are uniformly convergent 2nx fn (2) = 1+n*_z, 2 € [0, 1]
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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![Check if the following sequences are uniformly convergent
2nx
1+n¹², x € [0, 1]
Hint: You may consider following steps
• Let x = [0, 1].
Show that
fn (2)
=
lim fn (x)
n→∞
2nx
lim
n→∞ 1+n²x2
• Define f: [0, 1] → R, f (x) = 0.
• Make a conclusion about pointwise convergence of {fn (x)}.
= 0
• Use calculus (find the absolute maximum of fn (x) − f (x)| on [0, 1]) to show that, for all x = [0, 1],
-
|fn (x) - f (x)| ≤ fn
• Fix € > 0 and find N € N, such that, for all x = [0, 1],
|fn (x) - f (x)| < €
• Make a conclusion about the uniform convergence of {fn}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2F5ca75b45-077d-4cfe-a818-d26326cd71e8%2F7ygl6ms_processed.png&w=3840&q=75)
Transcribed Image Text:Check if the following sequences are uniformly convergent
2nx
1+n¹², x € [0, 1]
Hint: You may consider following steps
• Let x = [0, 1].
Show that
fn (2)
=
lim fn (x)
n→∞
2nx
lim
n→∞ 1+n²x2
• Define f: [0, 1] → R, f (x) = 0.
• Make a conclusion about pointwise convergence of {fn (x)}.
= 0
• Use calculus (find the absolute maximum of fn (x) − f (x)| on [0, 1]) to show that, for all x = [0, 1],
-
|fn (x) - f (x)| ≤ fn
• Fix € > 0 and find N € N, such that, for all x = [0, 1],
|fn (x) - f (x)| < €
• Make a conclusion about the uniform convergence of {fn}.
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