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...
Related questions
Question
![**Question**: Which of the following is an example of a series such that \(\sum a_n\) converges but \(\sum |a_n|\) diverges?
**Options**:
A. \(\sum \frac{(-1)^n}{n^3}\)
B. \(\sum \frac{(-1)^{n^{3}}}{\sqrt{n}}\)
C. \(\sum \frac{(-1)^n}{\sqrt[3]{n}}\)
D. \(\sum n^{-3}\)
---
### Explanation of the Problem
This problem is asking for an example of an alternating series that converges, but whose corresponding series of absolute values diverges. This touches on the concept of conditional and absolute convergence in series.
1. **Conditional Convergence**: A series \(\sum a_n\) is conditionally convergent if it converges but the series of absolute values \(\sum |a_n|\) diverges. This often occurs in alternating series.
2. **Absolute Convergence**: A series \(\sum a_n\) is absolutely convergent if \(\sum |a_n|\) converges.
---
### Series Analysis
To determine which series meets the criteria, let's analyze each option:
#### Option A:
\(\sum \frac{(-1)^n}{n^3}\)
This is an alternating series of the form \(\sum (-1)^n b_n\) where \(b_n = \frac{1}{n^3}\). For the alternating series test (Leibniz’s Test), we need:
- \(b_n\) is positive,
- \(b_n\) is decreasing,
- \(\lim_{n \to \infty} b_n = 0\).
All conditions are met because \(\frac{1}{n^3}\) is positive, decreasing, and approaches 0 as \(n\) approaches infinity. Hence, \(\sum \frac{(-1)^n}{n^3}\) converges.
For absolute convergence, we need to check \(\sum \left| \frac{(-1)^n}{n^3} \right| = \sum \frac{1}{n^3}\). Since \(\frac{1}{n^3}\) is a p-series with \(p = 3 > 1\), it converges.
#### Option B:
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01ab8b4d-9f7b-47d6-9013-d88fec313f5d%2F5acc5f27-3419-44d4-8040-00d20eefc198%2Fdt9nrd_processed.png&w=3840&q=75)
Transcribed Image Text:**Question**: Which of the following is an example of a series such that \(\sum a_n\) converges but \(\sum |a_n|\) diverges?
**Options**:
A. \(\sum \frac{(-1)^n}{n^3}\)
B. \(\sum \frac{(-1)^{n^{3}}}{\sqrt{n}}\)
C. \(\sum \frac{(-1)^n}{\sqrt[3]{n}}\)
D. \(\sum n^{-3}\)
---
### Explanation of the Problem
This problem is asking for an example of an alternating series that converges, but whose corresponding series of absolute values diverges. This touches on the concept of conditional and absolute convergence in series.
1. **Conditional Convergence**: A series \(\sum a_n\) is conditionally convergent if it converges but the series of absolute values \(\sum |a_n|\) diverges. This often occurs in alternating series.
2. **Absolute Convergence**: A series \(\sum a_n\) is absolutely convergent if \(\sum |a_n|\) converges.
---
### Series Analysis
To determine which series meets the criteria, let's analyze each option:
#### Option A:
\(\sum \frac{(-1)^n}{n^3}\)
This is an alternating series of the form \(\sum (-1)^n b_n\) where \(b_n = \frac{1}{n^3}\). For the alternating series test (Leibniz’s Test), we need:
- \(b_n\) is positive,
- \(b_n\) is decreasing,
- \(\lim_{n \to \infty} b_n = 0\).
All conditions are met because \(\frac{1}{n^3}\) is positive, decreasing, and approaches 0 as \(n\) approaches infinity. Hence, \(\sum \frac{(-1)^n}{n^3}\) converges.
For absolute convergence, we need to check \(\sum \left| \frac{(-1)^n}{n^3} \right| = \sum \frac{1}{n^3}\). Since \(\frac{1}{n^3}\) is a p-series with \(p = 3 > 1\), it converges.
#### Option B:
\
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning