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
### Problem Statement
**Prove or disprove that the series below converges:**
\[
\sum_{n=1}^{\infty} \left( -\frac{1}{n} \right)^{n-1}
\]
### Explanation
In mathematical analysis, it is essential to determine whether an infinite series converges or diverges. Convergence of a series means that the sum of its infinite terms approaches a finite limit. For a series of the form \(\sum_{n=1}^{\infty} a_n\), we often analyze its terms \(a_n\) to determine its behavior.
In this particular problem, we are asked to prove or disprove the convergence of the series:
\[
\sum_{n=1}^{\infty} \left( -\frac{1}{n} \right)^{n-1}
\]
### Approach to Solution
To determine if the series converges, we can use various convergence tests, such as the Ratio Test, Root Test, or Direct Comparison Test, depending on which is most appropriate for the given series.
#### Step-by-Step Analysis:
1. **Ratio Test:**
- The Ratio Test involves taking the limit of the absolute value of the ratio of consecutive terms. Formally, it states that a series \(\sum a_n\) converges if:
\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1
\]
- In this series, \(a_n = \left( -\frac{1}{n} \right)^{n-1}\)
We will apply the Ratio Test to this series to determine convergence or divergence.
2. Substituting \(a_n\) into the Ratio Test:
- Evaluate the following limit:
\[
\lim_{n \to \infty} \left| \frac{\left( -\frac{1}{n+1} \right)^{n}}{\left( -\frac{1}{n} \right)^{n-1}} \right|
\]
3. Simplify the limit expression and evaluate.
By following the steps above methodically, the convergence or divergence of the series can be established.
### Conclusion
After performing the necessary calculations and applying appropriate tests, the conclusion will](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42eb3f21-115e-47f2-b59c-753292f0c3d6%2Fd0613947-f92c-4ea6-985a-eab5463c2317%2Fec5zn2o_processed.png&w=3840&q=75)
Transcribed Image Text:## Series Convergence
### Problem Statement
**Prove or disprove that the series below converges:**
\[
\sum_{n=1}^{\infty} \left( -\frac{1}{n} \right)^{n-1}
\]
### Explanation
In mathematical analysis, it is essential to determine whether an infinite series converges or diverges. Convergence of a series means that the sum of its infinite terms approaches a finite limit. For a series of the form \(\sum_{n=1}^{\infty} a_n\), we often analyze its terms \(a_n\) to determine its behavior.
In this particular problem, we are asked to prove or disprove the convergence of the series:
\[
\sum_{n=1}^{\infty} \left( -\frac{1}{n} \right)^{n-1}
\]
### Approach to Solution
To determine if the series converges, we can use various convergence tests, such as the Ratio Test, Root Test, or Direct Comparison Test, depending on which is most appropriate for the given series.
#### Step-by-Step Analysis:
1. **Ratio Test:**
- The Ratio Test involves taking the limit of the absolute value of the ratio of consecutive terms. Formally, it states that a series \(\sum a_n\) converges if:
\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1
\]
- In this series, \(a_n = \left( -\frac{1}{n} \right)^{n-1}\)
We will apply the Ratio Test to this series to determine convergence or divergence.
2. Substituting \(a_n\) into the Ratio Test:
- Evaluate the following limit:
\[
\lim_{n \to \infty} \left| \frac{\left( -\frac{1}{n+1} \right)^{n}}{\left( -\frac{1}{n} \right)^{n-1}} \right|
\]
3. Simplify the limit expression and evaluate.
By following the steps above methodically, the convergence or divergence of the series can be established.
### Conclusion
After performing the necessary calculations and applying appropriate tests, the conclusion will
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