3. For each of the following sequences of real-valued functions on R, use the definition to show that { fn (x)}, converges pointwise to the specified f (x) on the given set I; then determine whether or not the convergence is uniform. Use the definition or its negation to justify your conclusions concerning uni- form convergence. 2x (a) {fn (x)}, ;f (x) = 0; I = [0, 1] 1+ nx (b) {fn (x)} cos nx }; f (x) = 0; I = [0, 1]
3. For each of the following sequences of real-valued functions on R, use the definition to show that { fn (x)}, converges pointwise to the specified f (x) on the given set I; then determine whether or not the convergence is uniform. Use the definition or its negation to justify your conclusions concerning uni- form convergence. 2x (a) {fn (x)}, ;f (x) = 0; I = [0, 1] 1+ nx (b) {fn (x)} cos nx }; f (x) = 0; I = [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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![3. For each of the following sequences of real-valued functions on R, use the
definition to show that { fn (x)} converges pointwise to the specified f (x)
on the given set I; then determine whether or not the convergence is uniform.
Use the definition or its negation to justify your conclusions concerning uni-
form convergence.
(a) {/n (x)}의1
2x
;f (x) = 0; I = [0, 1]
1+ nx
(b) {fn (x)}1
Cos nx
;f (x) = 0; I = [0, 1]
(c) {fn (x)}_1
n3x
;f (x) = 0; I = [0, 1]
1+n*x
(d) {Jn (x)}의1
positive fixed real number
n³x
1+ p4r2 ; f (x) = 0; I = [a, ) where a is a
(0) [S. 6) = {s(4) =
(1) (fu (x)}; = {nxe-* ; f (x) = 0; I = [0, 1]
:I =
-nx²
n=1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae257d6e-8d1d-4070-b937-ad737dc001ea%2F5f42490b-3cec-4cad-af3d-1459d0a9f828%2Fgf2e8rb_processed.gif&w=3840&q=75)
Transcribed Image Text:3. For each of the following sequences of real-valued functions on R, use the
definition to show that { fn (x)} converges pointwise to the specified f (x)
on the given set I; then determine whether or not the convergence is uniform.
Use the definition or its negation to justify your conclusions concerning uni-
form convergence.
(a) {/n (x)}의1
2x
;f (x) = 0; I = [0, 1]
1+ nx
(b) {fn (x)}1
Cos nx
;f (x) = 0; I = [0, 1]
(c) {fn (x)}_1
n3x
;f (x) = 0; I = [0, 1]
1+n*x
(d) {Jn (x)}의1
positive fixed real number
n³x
1+ p4r2 ; f (x) = 0; I = [a, ) where a is a
(0) [S. 6) = {s(4) =
(1) (fu (x)}; = {nxe-* ; f (x) = 0; I = [0, 1]
:I =
-nx²
n=1
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