Question

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} \sin ^{2} n\)

Short answer

The series diverges.

Step-by-step solution

Identify the Type of Series

The series is \[\sum_{n=1}^{\infty}(-1)^{n+1} \sin^{2} n\]This is an alternating series because of the factor \((-1)^{n+1}\).

Test for Absolute Convergence

Consider the absolute series \[\sum_{n=1}^{\infty} \left| (-1)^{n+1} \sin^{2} n \right| = \sum_{n=1}^{\infty} \sin^{2} n\]The function \(\sin^{2} n\) oscillates between 0 and 1, and does not tend to zero as \(n\) goes to infinity. Hence, the series does not converge absolutely.

Analyze Conditional Convergence

To check for conditional convergence using the Alternating Series Test, we need to see if \(\sin^{2} n\) tends to zero as \(n\) approaches infinity. Since \(\sin^{2} n\) continues to oscillate and does not approach zero, the Alternating Series Test fails.

Conclude the Type of Convergence

Given that the series does not converge absolutely (from Step 2) and does not converge conditionally (from Step 3), we conclude that the series diverges.

Key concepts

Alternating Series

An alternating series is a series in which the terms alternate in sign. A classic example is when each consecutive term is multiplied by \((-1)^{n+1}\). This makes the series switch between positive and negative terms, creating a zigzag pattern. For instance, our series \(\sum_{n=1}^{\infty}(-1)^{n+1} \sin^{2} n\) :: - starts with a positive term when \(n=1\)- becomes negative for \(n=2\)- and continues this pattern.This alternation is important in determining whether a series converges. The Alternating Series Test checks if:- the series terms decrease in absolute value- the limit of the terms as \(n\) approaches infinity is zero.If both conditions are met, the series converges conditionally. However, in this series, \( \sin^{2} n\) does not approach zero, so the series fails this test.

Absolute Convergence

Absolute convergence checks whether the series still converges when taking the absolute value of each term. This means any negative signs are ignored, and we only consider the magnitude of the terms. If a series converges absolutely, it converges under any condition, making it a strong form of convergence.For the series \(\sum_{n=1}^{\infty}(-1)^{n+1} \sin^{2} n\),if we consider its absolute series, it becomes:\[\sum_{n=1}^{\infty} \left| \sin^{2} n \right| = \sum_{n=1}^{\infty} \sin^{2} n\]Since \( \sin^{2} n\) fluctuates between 0 and 1 without tending towards zero as \( n \) goes to infinity, the series does not converge absolutely. A series that does not converge absolutely might still converge conditionally, but this isn't the case here.

Conditional Convergence

A series is conditionally convergent if it converges, but not absolutely. To test conditional convergence in an alternating series, the Alternating Series Test can be applied. It involves two key checks:

• The terms of the series should decrease in magnitude.
• The terms should go to zero as \(n\) goes to infinity.

Let's consider the series \(\sum_{n=1}^{\infty}(-1)^{n+1} \sin^{2} n\). Since \( \sin^{2} n\) continues to oscillate between 0 and 1, it does not trend toward zero. This means the series does not have conditional convergence. Conditional convergence is a softer convergence, different from the stronger absolute form. But here, neither occurs.

Divergence

The concept of divergence relates to series that fail to converge. A series diverges if the sum of its terms does not settle into a fixed number. In our case,\[\sum_{n=1}^{\infty}(-1)^{n+1} \sin^{2} n\]fails to meet the criteria for both absolute and conditional convergence. This can be explained by the lack of limits for \(\sin^{2} n\) as it never settles to zero, hence the series does not stabilize. Divergence tells us that adding more terms leads to a sum growing indefinitely, oscillating, or failing to approach a limit. Therefore, the series is said to diverge, meaning it simply doesn't add up to any meaningful number over its infinite extent.