Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{\sin n}{n \sqrt{n}} $$

Step-by-Step Solution

Answer

The series is conditionally convergent.

Step 1: Test for Convergence using Alternating Series Test

First, we note that the given series is an alternating series because of the term \((-1)^n\). The Alternating Series Test requires that \(b_n = \frac{\sin n}{n\sqrt{n}}\) is decreasing and \(\lim_{n \to \infty} b_n = 0\). Since \(|\sin n| \leq 1\), we have \(0 \leq \frac{|\sin n|}{n\sqrt{n}} \leq \frac{1}{n\sqrt{n}}\), which implies \(\lim_{n \to \infty} \frac{\sin n}{n\sqrt{n}} = 0\). This confirms convergence by the Alternating Series Test.

Step 2: Test for Absolute Convergence

To test for absolute convergence, consider \(\sum_{n=1}^{\infty} \left|(-1)^n \frac{\sin n}{n\sqrt{n}}\right| = \sum_{n=1}^{\infty} \frac{|\sin n|}{n\sqrt{n}}\). Since \(|\sin n| \leq 1\), the series \(\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}\) is a p-series with \(p = 3/2 > 1\), which is convergent. However, due to \(\sin n\) not being of constant sign, \(\sum_{n=1}^{\infty} \frac{|\sin n|}{n^{3/2}}\) does not actually converge. Therefore, the original series is not absolutely convergent.

Step 3: Conclusion on Convergence Type

Given that the series converges by the Alternating Series Test (Step 1) but does not converge absolutely (Step 2), we classify the series as conditionally convergent.

Key Concepts

Alternating Series

An alternating series is a series where the terms alternate in sign. The given series, \[\sum_{n=1}^{\infty}(-1)^{n} \frac{\sin n}{n \sqrt{n}},\]is an example of an alternating series due to the presence of the \[(-1)^n\] factor. This alternating sign creates a sequence where terms are subtracted and added alternately.

To determine if an alternating series converges, we often employ the **Alternating Series Test**. This test states that an alternating series converges if the absolute value of the terms \(b_n = \frac{\sin n}{n\sqrt{n}}\) satisfies two conditions: it must be decreasing and have a limit of zero as \(n\) goes to infinity.

In our example, the sine function's absolute value \(|\sin n| \leq 1\), does not increase the fraction \(\frac{\sin n}{n\sqrt{n}}\). Thus, it can be bounded and simplified to show that \(b_n\) approaches zero. Therefore, our series converges by the Alternating Series Test.

Absolute Convergence

Absolute convergence of a series involves checking whether the series of absolute values converges. If a series is absolutely convergent, it converges to its sum regardless of the signs of its terms.

For our given series, the testing for absolute convergence approaches it by considering:\[\sum_{n=1}^{\infty} \left|(-1)^n \frac{\sin n}{n \sqrt{n}}\right| = \sum_{n=1}^{\infty} \frac{|\sin n|}{n \sqrt{n}}\]We suspect the issue may arise because \(|\sin n|\) is not always positive, thus we reconsider it as a related p-series. The p-series \(\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}\) is known to converge as long as \(p > 1\).

In this test, it turns out that \(\sum_{n=1}^{\infty} \frac{|\sin n|}{n^{3/2}}\) doesn't converge absolutely due to the variability of \(\sin n\). This demonstrates that even if the general p-series converges, having a \(\sin n\) in there complicates things.

Conditional Convergence

Conditional convergence refers to a scenario where a series converges overall, but the series of absolute values diverges.

This is exactly the case with our given series:\[\sum_{n=1}^{\infty}(-1)^{n} \frac{\sin n}{n \sqrt{n}}\]It converges through the Alternating Series Test, as we previously established. However, it fails the absolute convergence test, thus making it an example of a conditionally convergent series.

Understanding conditional convergence is crucial. It shows how a series can be delicately balanced due to its alternating nature. Altering the order or adding terms can lead to different behaviors, unlike absolutely convergent series that remain steady under reordering.

This concept is vital in understanding the series behavior, especially in fields requiring approximation methods.