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

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

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 19E
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Need help with Task 4, this is a homework question not an exam
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 a 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. n=1 (x - 3)" 2n(n-1) Proofs Task 4 (portfolio). Let an be a sequence decreasing to zero. Prove that n=1(-1)"an converges. Task 5. Let an be a positive sequence that decreases to zero. Set Fall 2023 N SN = Σa an. n=1 Is lim sup Sy 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 and 3
Transcribed Image Text: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 a 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. n=1 (x - 3)" 2n(n-1) Proofs Task 4 (portfolio). Let an be a sequence decreasing to zero. Prove that n=1(-1)"an converges. Task 5. Let an be a positive sequence that decreases to zero. Set Fall 2023 N SN = Σa an. n=1 Is lim sup Sy 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 and 3
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