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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![**Problem 5: Determine if the Alternating Series Converges**
\[ \sum_{n=2}^{\infty} (-1)^n n \]
In this problem, you need to determine whether the given alternating series converges. The series is represented by the sum from \( n = 2 \) to infinity of \( (-1)^n n \).
### Explanation:
- **Alternating Series**: This is a series where the terms alternate in sign. Each term in the series is multiplied by \( (-1)^n \), which alternates between -1 and 1 as \( n \) changes from even to odd.
- **Convergence Criteria for Alternating Series**: For an alternating series \( \sum (-1)^n a_n \) to converge, the following conditions must be met:
1. The absolute value of the terms \( a_n \) must be decreasing: \( a_{n+1} \leq a_n \) for all \( n \).
2. The limit of the terms must approach zero: \( \lim_{n \to \infty} a_n = 0 \).
### Analysis:
In this case, \( a_n = n \), which does not approach zero as \( n \) approaches infinity. Additionally, the terms are not decreasing, as \( n \) increases with every successive term. Consequently, based on these criteria, the series does not converge.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe73d2f7-f48d-4799-ad26-f906d4f73fa5%2F4921bd3c-8e6e-47bd-91e7-3503d85846c4%2Fvaylqe8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 5: Determine if the Alternating Series Converges**
\[ \sum_{n=2}^{\infty} (-1)^n n \]
In this problem, you need to determine whether the given alternating series converges. The series is represented by the sum from \( n = 2 \) to infinity of \( (-1)^n n \).
### Explanation:
- **Alternating Series**: This is a series where the terms alternate in sign. Each term in the series is multiplied by \( (-1)^n \), which alternates between -1 and 1 as \( n \) changes from even to odd.
- **Convergence Criteria for Alternating Series**: For an alternating series \( \sum (-1)^n a_n \) to converge, the following conditions must be met:
1. The absolute value of the terms \( a_n \) must be decreasing: \( a_{n+1} \leq a_n \) for all \( n \).
2. The limit of the terms must approach zero: \( \lim_{n \to \infty} a_n = 0 \).
### Analysis:
In this case, \( a_n = n \), which does not approach zero as \( n \) approaches infinity. Additionally, the terms are not decreasing, as \( n \) increases with every successive term. Consequently, based on these criteria, the series does not converge.
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