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 Statement:**
(4) Determine if the alternating series converges
\[
\sum_{n=2}^{\infty} (-1)^n \sin\frac{1}{n}.
\]
**Explanation:**
This problem asks us to analyze the convergence of the given alternating series. The series is defined as the sum of terms from \( n = 2 \) to infinity, where each term is structured as \( (-1)^n \sin\frac{1}{n} \).
To determine convergence:
1. **Alternating Series Test**: This test can be applied to a series of the form \(\sum_{n=1}^{\infty} (-1)^n a_n\), where \(a_n > 0\).
- The terms \(a_n = \sin\frac{1}{n}\) should be positive and must converge to 0 as \(n\) approaches infinity.
- The sequence \(a_n\) should be non-increasing for sufficiently large \(n\).
2. **Examine the Term \(\sin\frac{1}{n}\)**:
- Since \(\sin x \approx x\) for small \(x\), \(\sin\frac{1}{n} \approx \frac{1}{n}\) as \(n\) grows.
- Thus, \(\sin\frac{1}{n}\) approaches 0 as \(n \to \infty\).
3. **Check if \(\{\sin\frac{1}{n}\}\) is Non-increasing**:
- We need to confirm that \(\sin\frac{1}{n+1} \leq \sin\frac{1}{n}\) for large \(n\).
- Since \(\frac{1}{n+1} < \frac{1}{n}\), and \(\sin x\) is an increasing function for \(x > 0\), this condition is satisfied.
If both conditions are met, the series converges by the Alternating Series Test.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe73d2f7-f48d-4799-ad26-f906d4f73fa5%2F1166854d-8d1b-4153-8e1c-cc1631e73aed%2Fyehun5e_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
(4) Determine if the alternating series converges
\[
\sum_{n=2}^{\infty} (-1)^n \sin\frac{1}{n}.
\]
**Explanation:**
This problem asks us to analyze the convergence of the given alternating series. The series is defined as the sum of terms from \( n = 2 \) to infinity, where each term is structured as \( (-1)^n \sin\frac{1}{n} \).
To determine convergence:
1. **Alternating Series Test**: This test can be applied to a series of the form \(\sum_{n=1}^{\infty} (-1)^n a_n\), where \(a_n > 0\).
- The terms \(a_n = \sin\frac{1}{n}\) should be positive and must converge to 0 as \(n\) approaches infinity.
- The sequence \(a_n\) should be non-increasing for sufficiently large \(n\).
2. **Examine the Term \(\sin\frac{1}{n}\)**:
- Since \(\sin x \approx x\) for small \(x\), \(\sin\frac{1}{n} \approx \frac{1}{n}\) as \(n\) grows.
- Thus, \(\sin\frac{1}{n}\) approaches 0 as \(n \to \infty\).
3. **Check if \(\{\sin\frac{1}{n}\}\) is Non-increasing**:
- We need to confirm that \(\sin\frac{1}{n+1} \leq \sin\frac{1}{n}\) for large \(n\).
- Since \(\frac{1}{n+1} < \frac{1}{n}\), and \(\sin x\) is an increasing function for \(x > 0\), this condition is satisfied.
If both conditions are met, the series converges by the Alternating Series Test.
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