Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
This is not a graded question and not an exam. This is a homework question that I need help. This is real analysis by the way.
MAT4405 Real Analysis Homework Number 4 Due: September 20, 2023 Meta Tasks Sorry this one is late, take until Monday if you want. Task 1. Prove that the sequence an converges. The sequence is defined by ao = √2 and an = √2+an-1. Task 2. Let an be a sequence such that every open interval containing 3 has infinitely many points from the sequence. Similarly every open interval containing -2 has infinitely many points from the sequence. For every x -2,3 there exists an open interval around x such that the interval only contains finitely many points from the sequence. Say as much as you can about this sequence. Task 3. Prove, including all relevant details, what interval the following series converges absolutely on. Check what happens at the endpoints. 1 • Every n=1 Proofs Task 4 (portfolio). Let an be a sequence decreasing to zero. Prove that Ex-1(-1)" an converges. Task 5. Let an be a positive sequence that decreases to zero. Set open (x - 3)" 2n(n-1) N SN=an. n=1 Is lim sup SN necessarily finite? Provide either a proof that it is or a counterexample that it isn't always. Task 6. Prove Proposition 1.3.10. Fall 2023 interval has -2 and 3
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Transcribed Image Text:MAT4405 Real Analysis Homework Number 4 Due: September 20, 2023 Meta Tasks Sorry this one is late, take until Monday if you want. Task 1. Prove that the sequence an converges. The sequence is defined by ao = √2 and an = √2+an-1. Task 2. Let an be a sequence such that every open interval containing 3 has infinitely many points from the sequence. Similarly every open interval containing -2 has infinitely many points from the sequence. For every x -2,3 there exists an open interval around x such that the interval only contains finitely many points from the sequence. Say as much as you can about this sequence. Task 3. Prove, including all relevant details, what interval the following series converges absolutely on. Check what happens at the endpoints. 1 • Every n=1 Proofs Task 4 (portfolio). Let an be a sequence decreasing to zero. Prove that Ex-1(-1)" an converges. Task 5. Let an be a positive sequence that decreases to zero. Set open (x - 3)" 2n(n-1) N SN=an. n=1 Is lim sup SN necessarily finite? Provide either a proof that it is or a counterexample that it isn't always. Task 6. Prove Proposition 1.3.10. Fall 2023 interval has -2 and 3
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