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. MAT4405Real AnalysisHomework Number 4Due: September 20, 2023Meta TasksSorry 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 thereexists 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 whathappens at the endpoints.1• Everyn=1ProofsTask 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. Setopen(x - 3)"2n(n-1)NSN=an.n=1Is 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 2023intervalhas -2 and 3

Answered: Image /qna-images/answer/a8a76ccb-9582-489a-bcbe-332eca0155b7.jpg

Answered: you want. 1. Prove that the sequence an…

you want. 1. Prove that the sequence an converges. The sequence is defined by ao = Task 2. Let an be a sequence such that every on Similarly every on √2 and an = √2+an-1.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 22E

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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

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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