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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![### Convergence and Divergence of Series
#### Problem Statement
**Question 4:** Determine if the following series converges or diverges. Show your work.
\[
\sum_{n=1}^{\infty} \frac{\sqrt[3]{n}}{n + 2}
\]
### Solution Approach
To solve this problem, we need to determine whether the given series converges or diverges. Here are several methods one can use to approach this problem, such as the comparison test, ratio test, root test, or integral test.
**Step-by-Step Solution:**
1. **Comparison with a Known Series:**
- Compare the given series with a known p-series or similar simplified series to determine convergence or divergence.
2. **Calculation and Limits:**
- Evaluate limits and determine the general behavior of the terms in the denominator and numerator as \( n \) approaches infinity.
### Additional Resources:
- **Comparison Test:** A comparison test is often used to compare a given series with a benchmark series to determine convergence or divergence.
- **Ratio/Test Root:** Other useful techniques can provide quicker convergence determinations for more complex series.
### Graphs/Diagrams:
There are no graphs or diagrams associated with this particular problem. However, one could provide a plot of the terms \(\frac{\sqrt[3]{n}}{n + 2}\) to visually inspect the behavior as \( n \to \infty \).
### Conclusion:
By showing detailed steps and using appropriate convergence tests, we can conclude whether the series converges or diverges. This aids in understanding the long-term behavior of infinite series, an essential concept in mathematical analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dbb4ae4-0d65-4baa-9481-63f79be91eca%2Fc8d8e4ca-ebfa-413a-ad76-6ccdefec252a%2Fd74vtw_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Convergence and Divergence of Series
#### Problem Statement
**Question 4:** Determine if the following series converges or diverges. Show your work.
\[
\sum_{n=1}^{\infty} \frac{\sqrt[3]{n}}{n + 2}
\]
### Solution Approach
To solve this problem, we need to determine whether the given series converges or diverges. Here are several methods one can use to approach this problem, such as the comparison test, ratio test, root test, or integral test.
**Step-by-Step Solution:**
1. **Comparison with a Known Series:**
- Compare the given series with a known p-series or similar simplified series to determine convergence or divergence.
2. **Calculation and Limits:**
- Evaluate limits and determine the general behavior of the terms in the denominator and numerator as \( n \) approaches infinity.
### Additional Resources:
- **Comparison Test:** A comparison test is often used to compare a given series with a benchmark series to determine convergence or divergence.
- **Ratio/Test Root:** Other useful techniques can provide quicker convergence determinations for more complex series.
### Graphs/Diagrams:
There are no graphs or diagrams associated with this particular problem. However, one could provide a plot of the terms \(\frac{\sqrt[3]{n}}{n + 2}\) to visually inspect the behavior as \( n \to \infty \).
### Conclusion:
By showing detailed steps and using appropriate convergence tests, we can conclude whether the series converges or diverges. This aids in understanding the long-term behavior of infinite series, an essential concept in mathematical analysis.
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