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4. Prove that 12 does not converge uniformly on S = [1,∞). Hint: Use Lemma 17.5 from Bartle and the pointwise limit in problem 3. n=1

4. Prove that 12 does not converge uniformly on S = [1,∞). Hint: Use Lemma 17.5 from Bartle and the pointwise limit in problem 3. n=1

Advanced Engineering Mathematics
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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Real Analysis II: ONLYQ4: Kindly find 2nd photo as guide for problem in Q4
3. Find the pointwise limit of 12 on S = [1, ∞). Hint: Define L E R n=1 n=1 by L = 1/2 (L = 2 but you don't need its exact value to do this problem.) 4. Prove that does not converge uniformly on S = [1, ∞). Hint: Use Lemma 17.5 from Bartle and the pointwise limit in problem 3.
Transcribed Image Text:3. Find the pointwise limit of 12 on S = [1, ∞). Hint: Define L E R n=1 n=1 by L = 1/2 (L = 2 but you don't need its exact value to do this problem.) 4. Prove that does not converge uniformly on S = [1, ∞). Hint: Use Lemma 17.5 from Bartle and the pointwise limit in problem 3.
17.5 LEMMA. A sequence (fn) does not converge uniformly on Do to f if and only if for some co>0 there is a subsequence (fm) of (fr) and a sequence (x) in Do such that (17,4) Ulf (1) f(₂ ||- for KEN.
Transcribed Image Text:17.5 LEMMA. A sequence (fn) does not converge uniformly on Do to f if and only if for some co>0 there is a subsequence (fm) of (fr) and a sequence (x) in Do such that (17,4) Ulf (1) f(₂ ||- for KEN.
Expert Solution
Step 1

let us consider the series by f_n(x)

fn(x) =∑1∞x2n2

we have∑1∞x2n2≤1n2

limn→∞fn(x)=f(x)=0

This implies that the series of function is converges pointwise to f(x)=0 on S=[1,inf)

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