Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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#22
Convergent? Why?
![### Convergence of Series and Sequences
#### Problem Set
**13.** Evaluate the convergence of the series:
\[
\sum_{n=1}^{\infty} \frac{2n}{n^2 + 1}
\]
**15.** Let \( a_n = \frac{2n}{3n + 1} \).
- (a) Determine whether the sequence \(\{a_n\}\) is convergent.
- (b) Determine whether the series \(\sum_{n=1}^{\infty} a_n\) is convergent.
**16.** Explain the difference between
- (a) \(\sum_{i=1}^{n} a_i\) and \(\sum_{j=1}^{n} a_j\)
- (b) \(\lim_{n \to \infty} a_i\) and \(\lim_{n \to \infty} a_j\)
**17–26.** Determine if the geometric series is convergent or divergent. If it is convergent, find its sum.
**17.** \(3 - 4 + \frac{16}{3} - \frac{64}{9} + \ldots\)
**18.** \(4 + 3 + \frac{9}{4} + \frac{27}{16} + \ldots\)
**19.** \(10 - 2 + 0.4 - 0.08 + \ldots\)
**20.** \(2 + 0.5 + 0.125 + 0.03125 + \ldots\)
**21.** \(\sum_{n=1}^{\infty} 12(0.73)^{n-1}\)
**22.** \(\sum_{n=1}^{\infty} \frac{5}{\pi^n}\)
**23.** \(\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{4^n}\)
**24.** \(\sum_{n=0}^{\infty} \frac{3^{n+1}}{(-2)^n}\)
**25.** \(\sum_{n=1}^{\infty} \frac{e^{2n}}{6^{n-1}}\)
**26.** \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2195755-482e-43b2-9535-9be7e3b0c44a%2Fb5708996-c52b-45f3-aa05-d651f509c3f2%2F2abxzzj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Convergence of Series and Sequences
#### Problem Set
**13.** Evaluate the convergence of the series:
\[
\sum_{n=1}^{\infty} \frac{2n}{n^2 + 1}
\]
**15.** Let \( a_n = \frac{2n}{3n + 1} \).
- (a) Determine whether the sequence \(\{a_n\}\) is convergent.
- (b) Determine whether the series \(\sum_{n=1}^{\infty} a_n\) is convergent.
**16.** Explain the difference between
- (a) \(\sum_{i=1}^{n} a_i\) and \(\sum_{j=1}^{n} a_j\)
- (b) \(\lim_{n \to \infty} a_i\) and \(\lim_{n \to \infty} a_j\)
**17–26.** Determine if the geometric series is convergent or divergent. If it is convergent, find its sum.
**17.** \(3 - 4 + \frac{16}{3} - \frac{64}{9} + \ldots\)
**18.** \(4 + 3 + \frac{9}{4} + \frac{27}{16} + \ldots\)
**19.** \(10 - 2 + 0.4 - 0.08 + \ldots\)
**20.** \(2 + 0.5 + 0.125 + 0.03125 + \ldots\)
**21.** \(\sum_{n=1}^{\infty} 12(0.73)^{n-1}\)
**22.** \(\sum_{n=1}^{\infty} \frac{5}{\pi^n}\)
**23.** \(\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{4^n}\)
**24.** \(\sum_{n=0}^{\infty} \frac{3^{n+1}}{(-2)^n}\)
**25.** \(\sum_{n=1}^{\infty} \frac{e^{2n}}{6^{n-1}}\)
**26.** \
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