integral test practice problems

Guri Bee

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18-10-2024

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Mastering the Integral Test: Practice Problems and Solutions

The integral test is a powerful tool in calculus for determining the convergence or divergence of infinite series. It allows us to relate the behavior of a series to the area under a corresponding continuous function. This article will explore several practice problems, providing step-by-step solutions and insightful explanations to solidify your understanding of the integral test.

Understanding the Integral Test

The integral test states that if a function $f(x)$ is positive, decreasing, and continuous on the interval $[1, \infty)$, then the infinite series $\sum_{n=1}^{\infty} f(n)$ converges if and only if the improper integral $\int_{1}^{\infty} f(x) \, dx$ converges.

Practice Problem 1: The Harmonic Series

Problem: Determine whether the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ converges or diverges using the integral test.

Solution:

1. Define the function: Let $f(x) = \frac{1}{x}$. This function is positive, decreasing, and continuous on the interval $[1, \infty)$.

2. Evaluate the improper integral:

   $\int_{1}^{\infty} \frac{1}{x} \, dx = \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x} \, dx = \lim_{b \to \infty} [\ln(x)]_{1}^{b} = \lim_{b \to \infty} (\ln(b) - \ln(1)) = \lim_{b \to \infty} \ln(b) = \infty$.

3. Conclusion: Since the improper integral diverges, the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ also diverges by the integral test.

Key Takeaway: The harmonic series is a classic example illustrating that even seemingly simple series can diverge.

Practice Problem 2: A Series with a Polynomial Term

Problem: Determine whether the series $\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$ converges or diverges using the integral test.

Solution:

1. Define the function: Let $f(x) = \frac{1}{x^2 + 1}$. This function is positive, decreasing, and continuous on the interval $[1, \infty)$.

2. Evaluate the improper integral:

   $\int_{1}^{\infty} \frac{1}{x^2 + 1} \, dx = \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^2 + 1} \, dx = \lim_{b \to \infty} [\arctan(x)]_{1}^{b} = \lim_{b \to \infty} (\arctan(b) - \arctan(1)) = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}$.

3. Conclusion: Since the improper integral converges, the series $\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$ also converges by the integral test.

Key Takeaway: The integral test is particularly useful for series involving rational functions, allowing us to leverage techniques from calculus to assess convergence.

Practice Problem 3: A Series with an Exponential Term

Problem: Determine whether the series $\sum_{n=1}^{\infty} \frac{e^{-n}}{n}$ converges or diverges using the integral test.

Solution:

1. Define the function: Let $f(x) = \frac{e^{-x}}{x}$. This function is positive, decreasing, and continuous on the interval $[1, \infty)$.

2. Evaluate the improper integral:

   $\int_{1}^{\infty} \frac{e^{-x}}{x} \, dx$ does not have a simple antiderivative and is not easily evaluated using elementary methods. However, we can employ the comparison test to show that the integral converges. Note that $\frac{e^{-x}}{x} < e^{-x}$ for all $x \geq 1$. Since $\int_{1}^{\infty} e^{-x} \, dx$ converges, the integral $\int_{1}^{\infty} \frac{e^{-x}}{x} \, dx$ also converges by the comparison test.

3. Conclusion: Since the improper integral converges, the series $\sum_{n=1}^{\infty} \frac{e^{-n}}{n}$ also converges by the integral test.

Key Takeaway: For more complex functions, we might need to rely on additional tests or techniques like the comparison test to determine convergence.

Additional Resources and Tips:

• For further practice, explore online resources like Khan Academy or MIT OpenCourseware.
• Remember to always verify that the conditions of the integral test are met before applying it to a series.
• Practice solving problems involving various types of functions, including polynomials, rational functions, and exponential functions.
• Utilize online integral calculators or symbolic math software to assist with evaluating complex integrals.

By diligently practicing and applying the integral test, you will gain a deeper understanding of infinite series and become adept at determining their convergence or divergence.