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. 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 =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.Σn=1Task 3. Prove, including all relevant details, what interval the following series converges absolutely on. Check whathappens at the endpoints.(x-3)"2n(n-1)Fall 20231.3.11√2 and an = √2+ an-1.NSN = Σan.n=1ProofsTask 4 (portfolio). Let an be a sequence decreasing to zero. Prove that En-1(-1)"an converges. partial sumTask 5. Let an be a positive sequence that decreases to zero. SetIs 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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Answered: Proofs Task 4 (portfolio). Let an be a…

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

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Question

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

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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