5. Prove that r € E' iff there exists a sequence (r„) in E\{r} such that r, → I.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need help with #5. Thank you.
![### Problem Set
1. **Prove or give counterexample.**
- (a) If \((x_n)\) and \((y_n)\) diverge, then \((x_n + y_n)\) diverges.
- (b) If \((x_n)\) and \((y_n)\) diverge, then \((x_n y_n)\) diverges.
- (c) If \((x_n)\) and \((x_n + y_n)\) converge, then \((y_n)\) converges.
- (d) If \((x_n)\) and \((x_n y_n)\) converge, then \((y_n)\) converges.
2. **Use the definition of convergence** to prove
\[
\lim_{n \to \infty} \frac{\cos n}{n^2 - n + 1} = 0.
\]
3. **Calculate the following limit analytically** using theorems from the notes.
\[
\lim_{n \to \infty} (\sqrt{n^2 + n} - n).
\]
4. **Let \(x_n \to 0\)** and \((y_n)\) be bounded. Prove \(x_n y_n \to 0\).
5. **Prove that \(x \in E'\) if and only if** there exists a sequence \((x_n)\) in \(E \setminus \{x\}\) such that \(x_n \to x\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F652eda4a-07c7-446c-baaa-e1ffddd9731f%2Fd5646821-012c-4aa5-9717-4be9c28c353f%2F4vvzwoh_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Set
1. **Prove or give counterexample.**
- (a) If \((x_n)\) and \((y_n)\) diverge, then \((x_n + y_n)\) diverges.
- (b) If \((x_n)\) and \((y_n)\) diverge, then \((x_n y_n)\) diverges.
- (c) If \((x_n)\) and \((x_n + y_n)\) converge, then \((y_n)\) converges.
- (d) If \((x_n)\) and \((x_n y_n)\) converge, then \((y_n)\) converges.
2. **Use the definition of convergence** to prove
\[
\lim_{n \to \infty} \frac{\cos n}{n^2 - n + 1} = 0.
\]
3. **Calculate the following limit analytically** using theorems from the notes.
\[
\lim_{n \to \infty} (\sqrt{n^2 + n} - n).
\]
4. **Let \(x_n \to 0\)** and \((y_n)\) be bounded. Prove \(x_n y_n \to 0\).
5. **Prove that \(x \in E'\) if and only if** there exists a sequence \((x_n)\) in \(E \setminus \{x\}\) such that \(x_n \to x\).
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