6. Consider the sequence of functionsn²a4 – sin(nx)fn(x)x € (0, 00)n2x(i) Show that fn converges pointwise and determine the limit.(ii) Use the inequality | sin x| < |x], x €R to show that fn converges uniformly to the above functionin (0, 00).(iii) Calculate the limitp² nx4 – sin(nx)limdx.n2xJustify any possible exchange of the integral with the limit.

Name: 6. Consider the sequence of functionsn²a4 – sin(nx)fn(x)x € (0, 00)n2
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Description: 6. Consider the sequence of functionsn²a4 – sin(nx)fn(x)x € (0, 00)n2x(i) Show that fn converges pointwise and determine the limit.(ii) Use the inequality | sin x| < |x], x €R to show that fn converges uniformly to the above functionin (0, 00).(iii)

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6. Consider the sequence of functions n²a* – sin(nx) fn(x) = x E (0, 00) n2x (i) Show that fn converges pointwise and determine the limit. (ii) Use the inequality | sin a| < |x\, x eR to show that fn converges uniformly to the above function in (0, 0). (iii) Calculate the limit na4 – sin(nx) lim dx. n 00 /1 n2x Justify any possible exchange of the integral with the limit.

6. Consider the sequence of functions n²a* – sin(nx) fn(x) = x E (0, 00) n2x (i) Show that fn converges pointwise and determine the limit. (ii) Use the inequality | sin a| < |x\, x eR to show that fn converges uniformly to the above function in (0, 0). (iii) Calculate the limit na4 – sin(nx) lim dx. n 00 /1 n2x Justify any possible exchange of the integral with the limit.

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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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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6. Consider the sequence of functions
n²a4 – sin(nx)
fn(x)
x € (0, 00)
n2x
(i) Show that fn converges pointwise and determine the limit.
(ii) Use the inequality | sin x| < |x], x €R to show that fn converges uniformly to the above function
in (0, 00).
(iii) Calculate the limit
p² nx4 – sin(nx)
lim
dx.
n2x
Justify any possible exchange of the integral with the limit.

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