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
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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}.
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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