Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The series is conditionally convergent.

Step 1: Determine Convergence

To start, we need to check if the given series \( \sum_{n=1}^{\infty}(-1)^{n+1} \sin \frac{\pi}{n} \) converges. We will use the alternating series test for this purpose. The alternating series test states that \( \sum_{n=1}^{\infty} (-1)^n a_n \) converges if two conditions are met: (1) \( a_n \to 0 \) as \( n \to \infty \), and (2) \( a_{n+1} \leq a_n \) for all \( n \).

Step 2: Apply Alternating Series Test

For \( \sum_{n=1}^{\infty}(-1)^{n+1} \sin \frac{\pi}{n} \), consider \( a_n = \sin \frac{\pi}{n} \). As \( n \to \infty \), \( \frac{\pi}{n} \to 0 \), so \( \sin \frac{\pi}{n} \to \sin 0 = 0 \). Thus, the first condition is satisfied. For the second condition, notice that \( \sin \frac{\pi}{n+1} < \sin \frac{\pi}{n} \) because the sine function is increasing on \( (0, \pi) \), which verifies the second condition. Hence, the series \( \sum_{n=1}^{\infty}(-1)^{n+1} \sin \frac{\pi}{n} \) is convergent by the alternating series test.

Step 3: Test for Absolute Convergence

Next, we need to check if the series is absolutely convergent. We do this by considering the series of absolute values: \( \sum_{n=1}^{\infty} \left| \sin \frac{\pi}{n} \right| = \sum_{n=1}^{\infty} \sin \frac{\pi}{n} \). Since \( \sin \frac{\pi}{n} \geq \frac{\pi}{n} \) for large \( n \), this series behaves like \( \sum_{n=1}^{\infty} \frac{\pi}{n} \), which is a harmonic series and is known to diverge. Hence, \( \sum_{n=1}^{\infty} \sin \frac{\pi}{n} \) diverges.

Step 4: Conclusion about Convergence

Since the series \( \sum_{n=1}^{\infty}(-1)^{n+1} \sin \frac{\pi}{n} \) converges, but its series of absolute values, \( \sum_{n=1}^{\infty} \sin \frac{\pi}{n} \), does not converge, the original series is conditionally convergent.

Key Concepts

Alternating Series Test

The Alternating Series Test is a handy tool in determining the convergence of series that have terms changing signs, such as:

• Positive, then negative, like in most alternating series.
• Negative, then positive, and so on.

For a series \ \( \sum_{n=1}^{\infty} (-1)^n a_n \ \), two key conditions need to be satisfied:

• First, \ \( a_n \to 0 \) as \ \( n \) approaches infinity. This means each term should become smaller and tend towards zero.
• Second, the terms \ \( a_{n+1} \leq a_n \) for all \ \( n \). This is the condition of decreasing terms in absolute value.

When these criteria are met, the series is confirmed to be convergent. Take for example the series \ \( \sum_{n=1}^{\infty}(-1)^{n+1} \sin \frac{\pi}{n} \ \). Here, the terms \ \( \sin \frac{\pi}{n} \to 0 \) as \ \( n \to \infty \), and they decrease as \ \( n \) grows. Thus, using the test, we confirm its convergence.

Absolute Convergence

Absolute convergence means a series converges even when you ignore the signs of the terms. To check for it, you take the absolute value of all terms:

• For example, evaluate \ \( \sum_{n=1}^{\infty} \left| \sin \frac{\pi}{n} \right| = \sum_{n=1}^{\infty} \sin \frac{\pi}{n} \ \).
• Here, we are checking if this series would converge without the alternating nature.

The series behaves similarly to the harmonic series \ \( \sum_{n=1}^{\infty} \frac{1}{n} \ \), which is known to diverge. Because the series of absolute values diverges, the original series cannot be absolutely convergent. In other words:

• If a series is not absolutely convergent, it means the behavior of alternating between positive and negative terms is crucial for its convergence.

Conditional Convergence

A series is conditionally convergent if it converges when considered normally, but not when you take absolute values of its terms. This often happens with alternating series:

• Conditionally convergent series have terms whose absolute series diverges, yet the full series converges.
• For \ \( \sum_{n=1}^{\infty}(-1)^{n+1} \sin \frac{\pi}{n} \ \), the actual series converges through the Alternating Series Test.
• However, the series of absolute values diverges because it mimics a harmonic series.

Understanding this distinction is important because it tells us that the pattern of sign changes plays a vital role in the convergence of such series. So, the original series is conditionally convergent because while its alternating structure ensures convergence, the lack of absolute convergence illustrates its dependence on negative terms.