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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![### Series Convergence Analysis
**Problem Statement:**
Determine whether the following series converge absolutely, converge conditionally, or diverge:
\[
\sum_{k=1}^{\infty} (-1)^k e^{-k}
\]
**Explanation:**
- The given mathematical expression is an infinite series starting from \(k = 1\) to infinity.
- Each term in the series is given by the general term \((-1)^k e^{-k}\).
**Steps to Determine Convergence:**
1. **Identify the Series Type:**
- The series is an alternating series because it has a factor of \((-1)^k\), which causes the signs of the terms to alternate.
2. **Check Absolute Convergence:**
- To check if the series converges absolutely, consider the absolute value of the terms:
\[
\left| (-1)^k e^{-k} \right| = e^{-k}
\]
- The series \(\sum_{k=1}^{\infty} e^{-k}\) is a geometric series with a common ratio \(r = e^{-1}\), which is less than 1.
- A geometric series converges if \(|r| < 1\).
3. **Check Conditional Convergence:**
- If the series converges absolutely, it also converges conditionally. However, if it does not converge absolutely, we should check for conditional convergence using the Alternating Series Test.
- The Alternating Series Test requires:
1. The absolute value of the terms \(e^{-k}\) must be monotonically decreasing.
2. The limit of \(e^{-k}\) as \(k\) approaches infinity must be 0.
- Both these conditions are satisfied for the series \(\sum_{k=1}^{\infty} (-1)^k e^{-k}\).
4. **Conclusion:**
- Since the series \(\sum_{k=1}^{\infty} e^{-k}\) converges (as a geometric series), the given series converges absolutely.
Therefore, the series \(\sum_{k=1}^{\infty} (-1)^k e^{-k}\) converges absolutely.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2ce945b-ebf8-47d5-81d6-18edb03d27a8%2F5ac8bee5-e3ff-4e2c-986f-0aa20c702800%2F9qx7f6s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Series Convergence Analysis
**Problem Statement:**
Determine whether the following series converge absolutely, converge conditionally, or diverge:
\[
\sum_{k=1}^{\infty} (-1)^k e^{-k}
\]
**Explanation:**
- The given mathematical expression is an infinite series starting from \(k = 1\) to infinity.
- Each term in the series is given by the general term \((-1)^k e^{-k}\).
**Steps to Determine Convergence:**
1. **Identify the Series Type:**
- The series is an alternating series because it has a factor of \((-1)^k\), which causes the signs of the terms to alternate.
2. **Check Absolute Convergence:**
- To check if the series converges absolutely, consider the absolute value of the terms:
\[
\left| (-1)^k e^{-k} \right| = e^{-k}
\]
- The series \(\sum_{k=1}^{\infty} e^{-k}\) is a geometric series with a common ratio \(r = e^{-1}\), which is less than 1.
- A geometric series converges if \(|r| < 1\).
3. **Check Conditional Convergence:**
- If the series converges absolutely, it also converges conditionally. However, if it does not converge absolutely, we should check for conditional convergence using the Alternating Series Test.
- The Alternating Series Test requires:
1. The absolute value of the terms \(e^{-k}\) must be monotonically decreasing.
2. The limit of \(e^{-k}\) as \(k\) approaches infinity must be 0.
- Both these conditions are satisfied for the series \(\sum_{k=1}^{\infty} (-1)^k e^{-k}\).
4. **Conclusion:**
- Since the series \(\sum_{k=1}^{\infty} e^{-k}\) converges (as a geometric series), the given series converges absolutely.
Therefore, the series \(\sum_{k=1}^{\infty} (-1)^k e^{-k}\) converges absolutely.
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