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Proofs Task 4 (portfolio). Let an be a sequence decreasing to zero. Prove that En-1(-1)"an converges. partial sum Task 5. Let a be a positive

Proofs Task 4 (portfolio). Let an be a sequence decreasing to zero. Prove that En-1(-1)"an converges. partial sum Task 5. Let a be a positive

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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I need help with Task 4, this is a homework question that I am stuck on and need help please. This is not an exam question or anything need help with it. This is real analysis.
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 = 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. Σ n=1 Task 3. Prove, including all relevant details, what interval the following series converges absolutely on. Check what happens at the endpoints. (x-3)" 2n(n-1) Fall 2023 1.3.11 √2 and an = √2+ an-1. N SN = Σan. n=1 Proofs Task 4 (portfolio). Let an be a sequence decreasing to zero. Prove that En-1(-1)"an converges. partial sum Task 5. Let an be a positive sequence that decreases to zero. Set 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. •Every open interval has -2 und 3
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 = 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. Σ n=1 Task 3. Prove, including all relevant details, what interval the following series converges absolutely on. Check what happens at the endpoints. (x-3)" 2n(n-1) Fall 2023 1.3.11 √2 and an = √2+ an-1. N SN = Σan. n=1 Proofs Task 4 (portfolio). Let an be a sequence decreasing to zero. Prove that En-1(-1)"an converges. partial sum Task 5. Let an be a positive sequence that decreases to zero. Set 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. •Every open interval has -2 und 3
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