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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Need help with Task 4, this is a homework question not an exam.
![**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 \( a_n \) converges. The sequence is defined by \( a_0 = \sqrt{2} \) and \( a_n = \sqrt{2 + a_{n-1}} \).
**Task 2.** Let \( a_n \) 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 \( x \neq -2, 3 \) 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.
\[
\sum_{n=1}^{\infty} \frac{(x-3)^n}{2n(n-1)}
\]
---
**Proofs**
**Task 4** (portfolio). Let \( a_n \) be a sequence decreasing to zero. Prove that \( \sum_{n=1}^{\infty} (-1)^n a_n \) converges.
**Task 5.** Let \( a_n \) be a positive sequence that decreases to zero. Set
\[
S_N = \sum_{n=1}^{N} a_n.
\]
Is \(\limsup S_N\) necessarily finite? Provide either a proof that it is or a counterexample that it isn’t always.
**Task 6.** Prove Proposition 1.3.10.
*Note: Handwritten note on the paper: "Every open interval has \(-2\) and 3".*
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Transcribed Image Text:**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 \( a_n \) converges. The sequence is defined by \( a_0 = \sqrt{2} \) and \( a_n = \sqrt{2 + a_{n-1}} \).
**Task 2.** Let \( a_n \) 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 \( x \neq -2, 3 \) 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.
\[
\sum_{n=1}^{\infty} \frac{(x-3)^n}{2n(n-1)}
\]
---
**Proofs**
**Task 4** (portfolio). Let \( a_n \) be a sequence decreasing to zero. Prove that \( \sum_{n=1}^{\infty} (-1)^n a_n \) converges.
**Task 5.** Let \( a_n \) be a positive sequence that decreases to zero. Set
\[
S_N = \sum_{n=1}^{N} a_n.
\]
Is \(\limsup S_N\) necessarily finite? Provide either a proof that it is or a counterexample that it isn’t always.
**Task 6.** Prove Proposition 1.3.10.
*Note: Handwritten note on the paper: "Every open interval has \(-2\) and 3".*
---
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