Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Using the Integral Test to Determine Series Convergence**
**Problem Statement:**
Use the Integral Test to determine if the following series converges. Be sure to prove that the Integral Test is valid for this series:
\[
\sum_{n=1}^{\infty} \frac{n}{(n^2+1)^2}
\]
**Explanation:**
The Integral Test can be used to determine the convergence of an infinite series. To apply the Integral Test, the function \( f(x) = \frac{x}{(x^2+1)^2} \) needs to be:
1. Positive
2. Continuous
3. Decreasing
on the interval \([1, \infty)\).
1. **Positivity:** The function is positive for \( x \geq 1 \).
2. **Continuity:** The function is continuous for all \( x \geq 1 \), since it's a rational function whose denominator does not equal zero in this interval.
3. **Decreasing Function Check:** To ensure the function is decreasing, compute the derivative \( f'(x) \) and evaluate whether it is negative for \( x \geq 1 \).
If these conditions are satisfied, compute the improper integral:
\[
\int_{1}^{\infty} \frac{x}{(x^2+1)^2} \, dx
\]
If the integral converges, so does the series. If the integral diverges, so does the series.](https://content.bartleby.com/qna-images/question/c1af4cad-3a82-431e-a6a5-54b2babd4110/63188370-9f70-4854-a167-3cedc9ccb711/xshx9w_processed.jpeg)
Transcribed Image Text:**Using the Integral Test to Determine Series Convergence**
**Problem Statement:**
Use the Integral Test to determine if the following series converges. Be sure to prove that the Integral Test is valid for this series:
\[
\sum_{n=1}^{\infty} \frac{n}{(n^2+1)^2}
\]
**Explanation:**
The Integral Test can be used to determine the convergence of an infinite series. To apply the Integral Test, the function \( f(x) = \frac{x}{(x^2+1)^2} \) needs to be:
1. Positive
2. Continuous
3. Decreasing
on the interval \([1, \infty)\).
1. **Positivity:** The function is positive for \( x \geq 1 \).
2. **Continuity:** The function is continuous for all \( x \geq 1 \), since it's a rational function whose denominator does not equal zero in this interval.
3. **Decreasing Function Check:** To ensure the function is decreasing, compute the derivative \( f'(x) \) and evaluate whether it is negative for \( x \geq 1 \).
If these conditions are satisfied, compute the improper integral:
\[
\int_{1}^{\infty} \frac{x}{(x^2+1)^2} \, dx
\]
If the integral converges, so does the series. If the integral diverges, so does the series.
Expert Solution
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Step 1
Given series,
Integral test:
Suppose that f(x) is a continuous, positive and decreasing function on the interval [n, ) and that fn=an then,
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