4. Consider the sequence of functions {fn}=2 where fn: [a, b] R defined for n > 2 by if 0 < x <- fn = {-n? (x -) if

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Question
Question 4
4. Consider the sequence of functions {f}-2 where fn: [a, b] R defined for n > 2 by
if 0 sxs
fn = {-n² (x-) if <xs.
if <x<1
n²x
72
0.
a. Prove {fn}2 converges pointwise to f(x) = 0.
b. Prove {f}-2 does not converge uniformly to f(x) = 0.
Show lim f (x)dx + lim A(x)dx by computing each integral where
C.
72- CO
graphing fn(x) may help.
5. Suppose {fn}n=1 and {gn-1 converges uniformly on E where {f-1 and {g,}-1 are
sequences of bounded functions. Prove that {fgn}, converges uniformly on E
Transcribed Image Text:4. Consider the sequence of functions {f}-2 where fn: [a, b] R defined for n > 2 by if 0 sxs fn = {-n² (x-) if <xs. if <x<1 n²x 72 0. a. Prove {fn}2 converges pointwise to f(x) = 0. b. Prove {f}-2 does not converge uniformly to f(x) = 0. Show lim f (x)dx + lim A(x)dx by computing each integral where C. 72- CO graphing fn(x) may help. 5. Suppose {fn}n=1 and {gn-1 converges uniformly on E where {f-1 and {g,}-1 are sequences of bounded functions. Prove that {fgn}, converges uniformly on E
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