Use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series. (-1)" Identify a, Evaluate the following limit. lim a Since lim a, O and a,+ a, for all n, the series is convergent ♥ n- 00
Use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series. (-1)" Identify a, Evaluate the following limit. lim a Since lim a, O and a,+ a, for all n, the series is convergent ♥ n- 00
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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![### Topic: Alternating Series Test for Convergence
**Problem Statement:**
Use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series:
\[
\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}
\]
**Tasks:**
1. **Identify \(a_n\):**
- Find the term \(a_n\) of the series. (Box left empty; marked with a red cross indicating incorrect or incomplete entry.)
2. **Evaluate the following limit:**
- \(\lim_{n \to \infty} a_n\) (Box left empty; marked with a red cross indicating incorrect or incomplete entry.)
3. **Conclusion:**
- Since \(\lim_{n \to \infty} a_n = 0\) (marked correct with a green check mark) and \(a_{n+1} \leq a_n\) for all \(n\) (inequality sign and conclusion verified as correct with green check marks for each), the series is convergent (marked correct with a green check mark).
This explanation guides the user through the process of identifying the necessary components for the Alternating Series Test, emphasizing the conditions for convergence: the limit of \(a_n\) must be zero, and the sequence \(a_n\) must be non-increasing.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F487e17f2-393e-4678-a513-01c89c0e6783%2F15441821-57aa-4e2f-bee7-63a135c387ea%2Fo6g46h_processed.png&w=3840&q=75)
Transcribed Image Text:### Topic: Alternating Series Test for Convergence
**Problem Statement:**
Use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series:
\[
\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}
\]
**Tasks:**
1. **Identify \(a_n\):**
- Find the term \(a_n\) of the series. (Box left empty; marked with a red cross indicating incorrect or incomplete entry.)
2. **Evaluate the following limit:**
- \(\lim_{n \to \infty} a_n\) (Box left empty; marked with a red cross indicating incorrect or incomplete entry.)
3. **Conclusion:**
- Since \(\lim_{n \to \infty} a_n = 0\) (marked correct with a green check mark) and \(a_{n+1} \leq a_n\) for all \(n\) (inequality sign and conclusion verified as correct with green check marks for each), the series is convergent (marked correct with a green check mark).
This explanation guides the user through the process of identifying the necessary components for the Alternating Series Test, emphasizing the conditions for convergence: the limit of \(a_n\) must be zero, and the sequence \(a_n\) must be non-increasing.
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