ofs (portfolio). Let an be a sequence decreasing to zero. Prove that En-1(-1)"an converges.
ofs (portfolio). Let an be a sequence decreasing to zero. Prove that En-1(-1)"an converges.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 22E
Related questions
Question
Need help with Task 4, this is a homework question not an exam.
![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.
Similarly every open interval containing -2 has infinitely many points from the sequence. For every x -2,3 there
Task 2. Let an be a sequence such that every open interval containing 3 has infinitely many points from the sequence.
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.
n=1
(x - 3)"
2n(n-1)
Fall 2023
Proofs
Task 4 (portfolio). Let an be a sequence decreasing to zero. Prove that En-1(-1)" an converges.
Task 5. Let an be a positive sequence that decreases to zero. Set
N
SN = Σα
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.
•Every open interval has -2 and 3
7
ex
ab
n
c](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47675d24-ba7f-4e21-93eb-97faaaed8f8d%2F2ac0af1d-c537-46df-a72b-077de1aaaac4%2Fdf3qx4_processed.jpeg&w=3840&q=75)
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.
Similarly every open interval containing -2 has infinitely many points from the sequence. For every x -2,3 there
Task 2. Let an be a sequence such that every open interval containing 3 has infinitely many points from the sequence.
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.
n=1
(x - 3)"
2n(n-1)
Fall 2023
Proofs
Task 4 (portfolio). Let an be a sequence decreasing to zero. Prove that En-1(-1)" an converges.
Task 5. Let an be a positive sequence that decreases to zero. Set
N
SN = Σα
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.
•Every open interval has -2 and 3
7
ex
ab
n
c
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