fn (x) = { 1 if x = 1, ¿, 4, 0 otherwise and let f be the pointwise limit of fn. Is each fn continuous at zero? Does fn + f uniformly on R? Is f continuous at zero? (b) Repeat this exercise using the sequence of functions * if r= 1,! 1 O otherwise. 1 In(x) = { *2' 3 .... (c) Repeat the exercise once more with the sequence 1 if r = ! if x = 1, 3, 3 0 otherwise. h„(x): 1 n-1 In each case, explain how the results are consistent with the content of the Continuous Limit Theorem (Theorem 6.2.6).

fn (x) = { 1 if x = 1, ¿, 4, 0 otherwise and let f be the pointwise limit of fn. Is each fn continuous at zero? Does fn + f uniformly on R? Is f continuous at zero? (b) Repeat this exercise using the sequence of functions * if r= 1,! 1 O otherwise. 1 In(x) = { *2' 3 .... (c) Repeat the exercise once more with the sequence 1 if r = ! if x = 1, 3, 3 0 otherwise. h„(x): 1 n-1 In each case, explain how the results are consistent with the content of the Continuous Limit Theorem (Theorem 6.2.6).

Name: (a) Define a sequence of functions on R by fn (x) = { 1 if x = 1, ¿, 4
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Description: (a) Define a sequence of functions on R by fn (x) = { 1 if x = 1, ¿, 4,0 otherwiseand let f be the pointwise limit of fn.Is each fn continuous at zero? Does fn + f uniformly on R? Is fcontinuous at zero?(b) Repeat this exercise using the sequence of

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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Question

(a) Define a sequence of functions on R by

fn (x) = { 1 if x = 1, ¿, 4,
0 otherwise
and let f be the pointwise limit of fn.
Is each fn continuous at zero? Does fn + f uniformly on R? Is f
continuous at zero?
(b) Repeat this exercise using the sequence of functions
* if r= 1,! 1
O otherwise.
1
In(x) = {
*2' 3 ....
(c) Repeat the exercise once more with the sequence
1 if r = !
if x = 1, 3, 3
0 otherwise.
h„(x):
1
n-1
In each case, explain how the results are consistent with the content of
the Continuous Limit Theorem (Theorem 6.2.6).

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