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(b) lim a,b, =-00 %3D

(b) lim a,b, =-00 %3D

Advanced Engineering Mathematics
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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#76 part b
## Convergence of Sequences and Series ### Section Problems: **Problem 76:** Suppose \(\lim_{n \to \infty} a_n = \infty\) and \(\lim_{n \to \infty} b_n = -\infty\) and \(\alpha \in \mathbb{R}\). Prove or give a counterexample: - (a) \(\lim_{n \to \infty} (a_n + b_n) = \infty\) - (b) \(\lim_{n \to \infty} (a_n b_n) = -\infty\) - (c) \(\lim_{n \to \infty} (\alpha a_n) = \infty\) - (d) \(\lim_{n \to \infty} (a_n b_n) = -\infty\) Finally, a sequence can diverge in other ways as the following problem displays. **Problem 77:** Show that each of the following sequences diverge. - (a) \(((-1)^n)_{n=1}^{\infty}\) - (b) \(((-1)^{n^2})_{n=1}^{\infty}\) - (c) \(a_n = \begin{cases} 1 & \text{if } n = 2^p \text{ for some } p \in \mathbb{N} \\ \frac{1}{n} & \text{otherwise} \end{cases}\) **Problem 78:** Suppose that \((a_n)_{n=1}^{\infty}\) diverges but not to infinity and that \(\alpha\) is a real number. What conditions on \(\alpha\) will guarantee that: - (a) \((\alpha a_n)_{n=1}^{\infty}\) converges? - (b) \((\alpha a_n)_{n=1}^{\infty}\) diverges? **Problem 79:** Show that if \(|r| > 1\) then \((r^n)_{n=1}^{\infty}\) diverges. Will it diverge to infinity? ### Additional Problems: **Problem 80:** Prove that if \(\lim_{n \
Transcribed Image Text:## Convergence of Sequences and Series ### Section Problems: **Problem 76:** Suppose \(\lim_{n \to \infty} a_n = \infty\) and \(\lim_{n \to \infty} b_n = -\infty\) and \(\alpha \in \mathbb{R}\). Prove or give a counterexample: - (a) \(\lim_{n \to \infty} (a_n + b_n) = \infty\) - (b) \(\lim_{n \to \infty} (a_n b_n) = -\infty\) - (c) \(\lim_{n \to \infty} (\alpha a_n) = \infty\) - (d) \(\lim_{n \to \infty} (a_n b_n) = -\infty\) Finally, a sequence can diverge in other ways as the following problem displays. **Problem 77:** Show that each of the following sequences diverge. - (a) \(((-1)^n)_{n=1}^{\infty}\) - (b) \(((-1)^{n^2})_{n=1}^{\infty}\) - (c) \(a_n = \begin{cases} 1 & \text{if } n = 2^p \text{ for some } p \in \mathbb{N} \\ \frac{1}{n} & \text{otherwise} \end{cases}\) **Problem 78:** Suppose that \((a_n)_{n=1}^{\infty}\) diverges but not to infinity and that \(\alpha\) is a real number. What conditions on \(\alpha\) will guarantee that: - (a) \((\alpha a_n)_{n=1}^{\infty}\) converges? - (b) \((\alpha a_n)_{n=1}^{\infty}\) diverges? **Problem 79:** Show that if \(|r| > 1\) then \((r^n)_{n=1}^{\infty}\) diverges. Will it diverge to infinity? ### Additional Problems: **Problem 80:** Prove that if \(\lim_{n \
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