Problem 38 - Calculus: Early Transcendentals

Question

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \( \displaystyle \sum_{n = 2}^{\infty} \frac {( - 1)^n}{n \ln n} \)

Step-by-Step Solution

Answer

The series is conditionally convergent.

Step 1: Identify the Series Type

The given series is \( \sum_{n=2}^{\infty} \frac{(-1)^n}{n \ln n} \), which is an alternating series because of the \((-1)^n\) factor. We will apply the Alternating Series Test and the Absolute Convergence Test to determine its convergence.

Step 2: Alternating Series Test

For the Alternating Series Test, we check if the sequence \( b_n = \frac{1}{n \ln n} \) is decreasing and if \( \lim_{n \to \infty} b_n = 0 \).1. Since \( n \ln n \) increases for \( n \geq 2 \), \( b_n \) decreases.2. \( \lim_{n \to \infty} \frac{1}{n \ln n} = 0 \).Since both conditions are satisfied, the original series is conditionally convergent.

Step 3: Absolute Convergence Test

For absolute convergence, consider the series \( \sum_{n=2}^{\infty} \left| \frac{(-1)^n}{n \ln n} \right| = \sum_{n=2}^{\infty} \frac{1}{n \ln n} \). We use the Integral Test to check convergence.Evaluate the integral \( \int_{2}^{\infty} \frac{1}{x \ln x} \, dx \).Make the substitution \( u = \ln x \), so \( du = \frac{1}{x} \, dx \). The integral becomes \( \int \frac{1}{u} \, du = \ln |u| + C = \ln \ln x + C \).The limit as \( x \to \infty \) of \( \ln \ln x \) approaches infinity, meaning the integral diverges. Hence, the series is not absolutely convergent.

Step 4: Conclusion

Since the series does not converge absolutely (the absolute version diverges) but satisfies the Alternating Series Test, the given series is conditionally convergent.

Key Concepts

Conditional Convergence

When we talk about conditional convergence, we are referring to a specific kind of behavior in series. A series is conditionally convergent when it converges, but does not converge absolutely. In simpler terms, the series

• converges when considered in its alternating form,
• but if you take the absolute values of its terms, the series diverges.

In our exercise, the series \[\sum_{n=2}^{\infty} \frac{(-1)^n}{n \ln n}\]is conditionally convergent. It met the conditions of the Alternating Series Test, indicating convergence in its alternating form. However, when we checked for absolute convergence, we discovered divergence. This shows that only the alternating form of this series behaves nicely in the realm of convergence.

Absolute Convergence

Absolute convergence is a stricter form of convergence than conditional convergence. It occurs when a series converges even when all terms are made positive. In other words, consider taking the absolute value of each term in the series:

• If this new series converges, the original series is also absolutely convergent.
• Absolute convergence implies conditional convergence, but the reverse is not true.

In the step-by-step solution, we examined the absolute convergence of the series by testing\[\sum_{n=2}^{\infty} \left| \frac{(-1)^n}{n \ln n} \right| = \sum_{n=2}^{\infty} \frac{1}{n \ln n}\]Using the Integral Test, we found this series diverges. Because of this divergence in its absolute form, the series is not absolutely convergent. Understanding this concept is key to determining the behavior of series in different scenarios.

Integral Test

The Integral Test is a handy tool for determining the convergence of series with positive terms. It involves comparing the series to an integral, which is easier to evaluate, to draw conclusions about the series' behavior.

• First, consider the function that corresponds to the terms of the series, ensuring it is continuous, positive, and decreasing.
• If the integral of this function from a point to infinity converges, so does the series.
• If the integral diverges, the series also diverges.

In our example,\[\int_{2}^{\infty} \frac{1}{x \ln x} \, dx\]was calculated and shown to diverge (\(\ln \ln x\) approaches infinity). Therefore, the series \(\sum_{n=2}^{\infty} \frac{1}{n \ln n}\) diverges, confirming that it's not absolutely convergent. The ability to use this test effectively helps in simplifying and understanding complex series.

Convergence of Series

Understanding the convergence of series is essential when dealing with infinite sums. Convergence tells us whether a sum approaches a finite value or not as more terms are added. Two important types of convergence are:

• Conditional Convergence: The series converges without needing the terms to all be positive, often through alternating terms.
• Absolute Convergence: If a series converges even after all terms are made positive, it has a stronger form of convergence.

Several tests, like the Alternating Series Test and Integral Test, help determine the convergence. These tests take different approaches but aim to simplify the problem by reducing it to a comparison with a known sum or behavior. Recognizing whether a series is conditionally or absolutely convergent helps predict its behavior in various mathematical contexts and applications.