Problem 24 - Calculus: Early Transcendentals

Question

Test the series for convergence or divergence. \( \displaystyle \sum_{n = 1}^{\infty} n \sin(1/n) \)

Step-by-Step Solution

Answer

The series diverges by the Test for Divergence.

Step 1: Define the Series

The series given is \( \sum_{n=1}^{\infty} n \sin\left(\frac{1}{n}\right) \). Our task is to determine whether this series converges or diverges. We will investigate the convergence using an appropriate convergence test.

Step 2: Simplify the Term

As \( n \to \infty \), the term \( \frac{1}{n} \to 0 \). Consequently, we use the approximation \( \sin\left(\frac{1}{n}\right) \approx \frac{1}{n} \) for small \( x \). Thus, the given term becomes approximately \( n \cdot \frac{1}{n} = 1 \).

Step 3: Evaluate the Behavior of the Series

The approximation \( a_n = 1 \) suggests that the terms do not approach zero as \( n \to \infty \). For a series \( \sum a_n \) to converge, the terms \( a_n \) must approach zero. Since \( a_n \approx 1 \), this condition is violated.

Step 4: Conclusion by the Test for Divergence

The Test for Divergence states that if the limit of \( a_n \) as \( n \to \infty \) is not zero, then the series \( \sum a_n \) diverges. In this series, \( \lim_{n \to \infty} n \sin\left(\frac{1}{n}\right) = 1 eq 0 \). Thus, the series diverges.

Key Concepts

Understanding the Test for Divergence

The Test for Divergence is a fundamental tool in determining whether a series \( \sum a_n \) converges or diverges. It is based on the behavior of the terms as \( n \to \infty \). To apply this test effectively, review the terms of your series.

• If \( \lim_{n \to \infty} a_n eq 0 \), the series diverges.
• If \( \lim_{n \to \infty} a_n = 0 \), this alone does not imply convergence, but it’s a necessary condition.

For our series, we simplify the term to find \( a_n = 1 \). Observing that \( \lim_{n \to \infty} a_n = 1 \), the series does not satisfy the condition for convergence, confirming its divergence.

Exploring Series Approximation

When dealing with series, particularly those involving functions like sine, approximation can be invaluable for simplification and understanding. The Taylor expansion for \( \sin(x) \) approximates \( \sin(x) \) as \( x \) nears zero. For small values of \( x \):

• \( \sin(x) \approx x \).

In our series \( \sum n \sin\left(\frac{1}{n}\right) \), the term becomes \( n \cdot \frac{1}{n} = 1 \), allowing us to approximate the series by constant terms, which is pivotal in applying the Test for Divergence. This approximation simplifies complex trigonometric expressions, making convergence tests more straightforward.

Understanding Sinusoidal Functions in Series

Sinusoidal functions like \( \sin(x) \) regularly appear in series, often requiring specific handling due to their oscillatory nature. As trigonometric functions, they offer unique behaviors near zero. Some core properties:

• \( \sin(x) \) is bounded between -1 and 1, oscillating as it goes.
• Its approximation close to zero: \( \sin(x) \approx x \).

These properties are essential for series involving sine, enabling approximation techniques. This behavior is leveraged for simplification into manageable terms within series involving \( \sin(\frac{1}{n}) \), as seen in the original exercise. Awareness of these traits aids in predicting the series' convergence or divergence.

The Importance of Limits in Series Analysis

The concept of limits is central in sequences and series, guiding us towards understanding long-term behavior. Calculating the limit involves evaluating how a function's value approaches a particular point as its input approaches a given figure—often infinity for series.For a series \( \sum a_n \),

• If \( \lim_{n \to \infty} a_n = 0 \), it suggests potential for convergence but doesn't guarantee it.
• If non-zero, divergence is confirmed.

Applying this to our series, discovering that \( \lim_{n \to \infty} n \sin\left(\frac{1}{n}\right) = 1 \) is crucial, directly leading to the decision of divergence by the Test for Divergence. Thus, understanding limits lets us draw conclusions regarding the infinite behavior of sequences and series.