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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Question
Show that the series below is conditionally convergent. Justify.
![The image depicts a mathematical series represented as:
\[
\sum_{n=2}^{\infty} \frac{(-1)^n}{n - 1}
\]
This is an infinite series that starts at \( n = 2 \) and continues indefinitely. Each term in the series is given by the formula:
\[
\frac{(-1)^n}{n - 1}
\]
### Explanation:
- **Summation Symbol (\(\sum\))**: This indicates that you are summing multiple terms from \( n = 2 \) to infinity.
- **\((-1)^n\)**: This part of the formula alternates the sign of each term. When \( n \) is even, \((-1)^n = 1\); when \( n \) is odd, \((-1)^n = -1\).
- **\(n - 1\)**: In the denominator, \( n - 1 \) implies that each term in the series is divided by one less than its index.
This series is an example from mathematical analysis involving alternating series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a3bdd75-f939-43d0-9bcd-1717dd46199e%2F5f68dc0f-0787-478f-89c8-08daa49fca9d%2Ftn7fbfs_processed.png&w=3840&q=75)
Transcribed Image Text:The image depicts a mathematical series represented as:
\[
\sum_{n=2}^{\infty} \frac{(-1)^n}{n - 1}
\]
This is an infinite series that starts at \( n = 2 \) and continues indefinitely. Each term in the series is given by the formula:
\[
\frac{(-1)^n}{n - 1}
\]
### Explanation:
- **Summation Symbol (\(\sum\))**: This indicates that you are summing multiple terms from \( n = 2 \) to infinity.
- **\((-1)^n\)**: This part of the formula alternates the sign of each term. When \( n \) is even, \((-1)^n = 1\); when \( n \) is odd, \((-1)^n = -1\).
- **\(n - 1\)**: In the denominator, \( n - 1 \) implies that each term in the series is divided by one less than its index.
This series is an example from mathematical analysis involving alternating series.
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