**Topic: Using the Integral Test to Determine Convergence of a Series**### Problem Statement:Use the integral test to determine whether the series \[\sum_{k=1}^{\infty} \frac{k^2}{(k^3 + 9)^2} \]converges.### Explanation:The integral test can be applied to determine the convergence of an infinite series by comparing it to an improper integral. To utilize the integral test, follow these steps:1. **Define the function**: Let \( f(x) = \frac{x^2}{(x^3 + 9)^2} \). 2. **Verify the conditions**: Ensure that \( f(x) \) is positive, continuous, and decreasing for \( x \ge 1 \).3. **Set up the integral**: Evaluate the improper integral from 1 to infinity: \[ \int_{1}^{\infty} \frac{x^2}{(x^3 + 9)^2} \, dx \]4. **Evaluate the integral**: This step involves detailed calculus computations to evaluate the improper integral: - Substitute \( u = x^3 + 9 \), hence \( du = 3x^2 \, dx \). - Adjust the bounds and evaluate the resulting integral.5. **Determine convergence**: If the improper integral converges, the series converges. If the integral diverges, the series diverges.By evaluating the integral, we determine the behavior of the series.### Graphs and Diagrams:There are no graphs or diagrams in this specific problem. However, to aid understanding, you can imagine plotting the function \( f(x) = \frac{x^2}{(x^3 + 9)^2} \) and observing its behavior as \( x \) increases. The function should decrease and approach zero as \( x \) flows towards infinity, reinforcing its suitability for the integral test.For a comprehensive solution, a step-by-step evaluation of the integral, consideration of every detail in the substitution, and simplification should be included in the full analysis process.

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Answered: k2 Use the integral test to determine…

k2 Use the integral test to determine whether 2 1319)2 converges. (k3 + 9)2 k=1

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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