**Topic: Series Convergence or Divergence** In this exercise, we aim to determine whether the given series converges or diverges. The series in question is presented as follows: \[ \sum_{n=1}^{\infty} \frac{1}{n-1/n} \] **Explanation:** - The symbol \(\sum\) denotes a summation. - \(n=1\) at the bottom of the summation is the starting point of the series. - \(\infty\) at the top of the summation indicates that the series continues indefinitely. - The general term of the series is \(\frac{1}{n - \frac{1}{n}}\). You are required to analyze this series to determine if it converges or diverges. This involves applying various mathematical tests and concepts related to infinite series. For instance: 1. **Simplifying the general term**: Start by simplifying \(\frac{1}{n - \frac{1}{n}}\). 2. **Comparison Test**: Compare the series with a known convergent or divergent series. 3. **Ratio Test or Root Test**: Apply these tests to check the behavior of the series as \(n\) approaches infinity. 4. **Integral Test**: Evaluate the integral of the function to determine convergence. By taking these steps, you will be able to conclude whether the series converges or diverges. For more hints and a detailed solution, please refer to the Math section under "Series and Sequences" on our website.
**Topic: Series Convergence or Divergence** In this exercise, we aim to determine whether the given series converges or diverges. The series in question is presented as follows: \[ \sum_{n=1}^{\infty} \frac{1}{n-1/n} \] **Explanation:** - The symbol \(\sum\) denotes a summation. - \(n=1\) at the bottom of the summation is the starting point of the series. - \(\infty\) at the top of the summation indicates that the series continues indefinitely. - The general term of the series is \(\frac{1}{n - \frac{1}{n}}\). You are required to analyze this series to determine if it converges or diverges. This involves applying various mathematical tests and concepts related to infinite series. For instance: 1. **Simplifying the general term**: Start by simplifying \(\frac{1}{n - \frac{1}{n}}\). 2. **Comparison Test**: Compare the series with a known convergent or divergent series. 3. **Ratio Test or Root Test**: Apply these tests to check the behavior of the series as \(n\) approaches infinity. 4. **Integral Test**: Evaluate the integral of the function to determine convergence. By taking these steps, you will be able to conclude whether the series converges or diverges. For more hints and a detailed solution, please refer to the Math section under "Series and Sequences" on our website.
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: Series Convergence or Divergence**
In this exercise, we aim to determine whether the given series converges or diverges.
The series in question is presented as follows:
\[ \sum_{n=1}^{\infty} \frac{1}{n-1/n} \]
**Explanation:**
- The symbol \(\sum\) denotes a summation.
- \(n=1\) at the bottom of the summation is the starting point of the series.
- \(\infty\) at the top of the summation indicates that the series continues indefinitely.
- The general term of the series is \(\frac{1}{n - \frac{1}{n}}\).
You are required to analyze this series to determine if it converges or diverges. This involves applying various mathematical tests and concepts related to infinite series. For instance:
1. **Simplifying the general term**: Start by simplifying \(\frac{1}{n - \frac{1}{n}}\).
2. **Comparison Test**: Compare the series with a known convergent or divergent series.
3. **Ratio Test or Root Test**: Apply these tests to check the behavior of the series as \(n\) approaches infinity.
4. **Integral Test**: Evaluate the integral of the function to determine convergence.
By taking these steps, you will be able to conclude whether the series converges or diverges.
For more hints and a detailed solution, please refer to the Math section under "Series and Sequences" on our website.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7bb6f29c-bae7-4b63-bf8e-21de0c88ea98%2F6893320d-3374-4d64-b5d8-f87226ad5aa7%2Fazhd91f.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Series Convergence or Divergence**
In this exercise, we aim to determine whether the given series converges or diverges.
The series in question is presented as follows:
\[ \sum_{n=1}^{\infty} \frac{1}{n-1/n} \]
**Explanation:**
- The symbol \(\sum\) denotes a summation.
- \(n=1\) at the bottom of the summation is the starting point of the series.
- \(\infty\) at the top of the summation indicates that the series continues indefinitely.
- The general term of the series is \(\frac{1}{n - \frac{1}{n}}\).
You are required to analyze this series to determine if it converges or diverges. This involves applying various mathematical tests and concepts related to infinite series. For instance:
1. **Simplifying the general term**: Start by simplifying \(\frac{1}{n - \frac{1}{n}}\).
2. **Comparison Test**: Compare the series with a known convergent or divergent series.
3. **Ratio Test or Root Test**: Apply these tests to check the behavior of the series as \(n\) approaches infinity.
4. **Integral Test**: Evaluate the integral of the function to determine convergence.
By taking these steps, you will be able to conclude whether the series converges or diverges.
For more hints and a detailed solution, please refer to the Math section under "Series and Sequences" on our website.
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