**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. Positive2. Continuous3. Decreasingon 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.

Answered: Use the Integral Test to detcrmine if…

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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**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

Step 1

Given series,

$\sum_{n = 1}^{\infty} \frac{n}{{(n^{2} + 1)}^{2}}$

Integral test:

Suppose that f(x) is a continuous, positive and decreasing function on the interval [n, $\infty$ ) and that f_n=a_n then,

$if \int_{k}^{\infty} f (x) dx is convergent so is \sum_{n = k}^{\infty} a_{n} . if \int_{k}^{\infty} f (x) dx is divergent so is \sum_{n = k}^{\infty} a_{n} .$

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