Question

In the following exercises, use an appropriate test to determine whether the series converges. $$ a_{k}=2^{\sin (1 / k)} $$

Short answer

The series diverges by the divergence test.

Step-by-step solution

Understand the Series

We are given the series with terms \( a_k = 2^{\sin(1/k)} \). Our task is to determine if this series converges or not. Applying the convergence test will help determine this.

Analyze the Series Behavior

Examine the expression \( 2^{\sin(1/k)} \). As \( k \to \infty \), \( \frac{1}{k} \to 0 \). The sine function, \( \sin(x) \), is continuous, so \( \sin(1/k) \to \sin(0) = 0 \). Thus, \( a_k = 2^{\sin(1/k)} \to 2^0 = 1 \).

Apply the Divergence Test

The divergence test (or test for divergence) states if \( \lim_{k \to \infty} a_k eq 0 \), then the series \( \sum a_k \) diverges. From Step 2, we found \( \lim_{k \to \infty} a_k = 1 \), which is not 0.

Conclude the Convergence Outcome

Since \( \lim_{k \to \infty} a_k = 1 eq 0 \), by the divergence test, the series \( \sum_{k=1}^{\infty} a_k \) diverges.

Key concepts

Convergence Tests

Convergence tests are fundamental tools in analyzing the behavior of infinite series to determine if they converge or diverge. These tests help us understand whether the sum of an infinite sequence of terms results in a finite number. Several convergence tests can be employed, each suitable for different types of series:

• The Ratio Test: Useful for series with factorials or exponential terms, where term comparison involves taking limits.
• The Root Test: It is similar to the ratio test but often easier for series involving power expressions.
• The Integral Test: Applicable when the series terms come from a function that can be integrated over an interval.
• The Comparison Test: Compares a series with a second series that is known to converge or diverge.

These tests are applied based on the characteristics of the series given. For the series \(a_k = 2^{\sin(1/k)}\), examining the term behavior leads to applying the divergence test as it fits the scenario where the limit of the terms doesn’t equal zero, indicating divergence without further tests needed.

Divergence Test

The divergence test is an important tool that provides a quick check on the convergence of a series. It states that if the limit of the terms of the series, \(a_k\), as \(k\) approaches infinity is not zero, then the series diverges. This is because a necessary condition for the convergence of an infinite series \(\sum a_k\) is that the terms themselves must approach zero as \(k\) grows larger.
Applying the divergence test is typically straightforward:

• Find \(\lim_{k \to \infty} a_k\).
• If the limit is not zero, the series \(\sum a_k\) definitely diverges.

In our example series \(a_k = 2^{\sin(1/k)}\), we calculated \(\lim_{k \to \infty} a_k = 1\) inside the solution. Since this limit isn't zero, the divergence test confirms that the series diverges without needing any further calculations.

Limit of a Sequence

The limit of a sequence is a key concept when analyzing series. It refers to the value that the terms of a sequence \(a_k\) approach as \(k\) becomes infinitely large. Understanding sequence limits is crucial because it lays the foundation for applying convergence and divergence tests.
For sequence limits, the steps include:

• Determine the behavior of the individual sequence \(a_k\) as \(k\) increases.
• Analyze if the terms approach a specific value.

In the exercise's context, the behavior of \(a_k = 2^{\sin(1/k)}\) is assessed using the limit \(\lim_{k \to \infty} a_k = 1\). This derives from \(\sin(1/k)\) heading towards zero as \(k\) grows, converting the expression to \(2^0 = 1\). Understanding this aspect allows students to apply the appropriate convergence test, such as the divergence test, when terms don't trend towards zero.