Question

Using the Integral Test For each of the following series, use the integral test to determine whether the series converges or diverges a. \(\sum_{n=1}^{\infty} 1 / n^{3}\)

Short answer

The series \(\sum_{n=1}^{\infty} \frac{1}{n^3}\) converges.

Step-by-step solution

Identify the Function

We start by identifying the function within the series. For the series \(\sum_{n=1}^{\infty} \frac{1}{n^3}\), the corresponding function is \(f(x) = \frac{1}{x^3}\), which is a positive, continuous, and decreasing function for \(x \geq 1\).

Set Up the Integral

Convert the series into an improper integral. The integral that corresponds to the series is \(\int_{1}^{\infty} \frac{1}{x^3}\, dx\).

Evaluate the Integral

Compute the improper integral \(\int_{1}^{\infty} \frac{1}{x^3}\, dx\). The antiderivative of \(\frac{1}{x^3}\) is \(-\frac{1}{2x^2}\). Evaluate it from 1 to \(\infty\).\[\lim_{{b \to \infty}} \left(-\frac{1}{2b^2}\right) + \frac{1}{2(1)^2} = 0 + \frac{1}{2} = \frac{1}{2}\].

Apply the Integral Test Conclusion

Since the improper integral \(\int_{1}^{\infty} \frac{1}{x^3}\, dx\) converges to \(\frac{1}{2}\), by the Integral Test, the series \(\sum_{n=1}^{\infty} \frac{1}{n^3}\) also converges.

Key concepts

Convergence of Series

When working with infinite series, one of the primary goals is to determine whether the series converges or diverges. Convergence of a series means that as you add more terms, the total sum approaches a finite value. For a series like \( \sum_{n=1}^{\infty} \frac{1}{n^3} \), convergence can be tested using various methods, one of which is the Integral Test.

The Integral Test is particularly useful for series that are positive and decreasing. This involves comparing the series to an improper integral. If the corresponding improper integral converges to a finite number, then the series also converges. Conversely, if the integral diverges, so does the series. In our exercise, after testing with the improper integral, we found that the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \) converges as the integral equals \( \frac{1}{2} \).

Convergence of series is an essential concept as it allows mathematicians to handle infinite processes with certainty, providing insights into stability and predictability in mathematical models.

Improper Integrals

Improper integrals form a crucial part of calculus, especially when determining the behavior of functions over infinite intervals. An integral is termed "improper" when it has either or both of its limits extending to infinity or when the integrand approaches infinity within the interval of integration.

For the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \), the corresponding improper integral is \( \int_{1}^{\infty} \frac{1}{x^3} \, dx \). Evaluating the integral involves calculating the antiderivative, which we found as \(-\frac{1}{2x^2} \), and assessing the limit as the upper bound approaches infinity.

Improper integrals are routinely used in calculus due to their relevance in characterizing and comparing the vast, often unmanageable bounds in mathematical analysis. Mastery of improper integrals allows students to determine the sum behavior over limitless extents.

Calculus Techniques

Calculus offers an array of techniques essential for evaluating series and integrals. One of these techniques is finding antiderivatives, an indispensable tool in computing integrals. In our example, the antiderivative of \( \frac{1}{x^3} \) was calculated as \(-\frac{1}{2x^2} \). This step is critical for determining the result of an improper integral.

Another key technique involves taking limits, especially when dealing with bounds going towards infinity, as seen in improper integrals. Calculating \( \lim_{{b \to \infty}} \left(-\frac{1}{2b^2}\right) + \frac{1}{2(1)^2} \) was vital for evaluating the integral and concluding the convergence of the series.

These calculus techniques, including evaluating limits and antiderivatives, offer students the tools they need to handle integrals and series thoroughly, enabling them to analyze and solve complex mathematical problems effectively. Developing proficiency in these techniques is crucial for anyone exploring advanced mathematical concepts.