Question

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

Short answer

The series diverges.

Step-by-step solution

Identify the Series Type

The given series is \(\sum_{n=1}^{\infty}(-1)^{n+1} \cos ^{2}(1 / n)\). It appears to be an alternating series because of the term \((-1)^{n+1}\).

Check for Absolute Convergence

To determine absolute convergence, consider the series \(\sum_{n=1}^{\infty} \left|\cos^{2}(1/n)\right| = \sum_{n=1}^{\infty} \cos^{2}(1/n)\). As \(n\) approaches infinity, \(\cos^{2}(1/n)\) approaches \(\cos^{2}(0) = 1\), thus the terms do not approach 0, meaning the series does not converge absolutely.

Apply Alternating Series Test for Conditional Convergence

For an alternating series \(\sum_{n=1}^{\infty} (-1)^{n+1} a_n\) to converge, \(a_n\) should decrease monotonically and \(\lim_{n \to \infty} a_n = 0\). Here, \(a_n = \cos^{2}(1/n)\). As \(n\to\infty, \cos^{2}(1/n)\to \cos^{2}(0) = 1\), so the limit is not zero. Hence, the series does not satisfy the conditions for conditional convergence either.

Conclude on Convergence

Since the series does not converge absolutely and does not meet the criteria for an alternating series to converge conditionally, the series diverges.

Key concepts

Absolute Convergence

In mathematical analysis, absolute convergence of a series is a stronger form of convergence. For a series \(\sum_{n=1}^{\infty} a_n\), it converges absolutely if the series of absolute values \(\sum_{n=1}^{\infty} |a_n|\) also converges. This means that if you turn every term into its positive value and the series still converges, then the original series is absolutely convergent.

Applying this concept to the series \(\sum_{n=1}^{\infty} (-1)^{n+1} \cos^2(1/n)\), we examine \(\sum_{n=1}^{\infty} \cos^2(1/n)\). As \(n\) becomes very large, \(1/n\) approaches zero and \(\cos^2(1/n)\) tends towards \(\cos^2(0) = 1\). Therefore, the terms do not get smaller and thus the series of absolute values does not converge.

Absolute convergence is an important concept because it guarantees the convergence of rearranged series, which is not always the case for conditionally convergent or divergent series.

Conditional Convergence

Conditional convergence occurs when a series converges, but it does not converge absolutely. In simpler terms, the series \(\sum_{n=1}^{\infty} a_n\) is conditionally convergent if it converges, but \(\sum_{n=1}^{\infty} |a_n|\) diverges. This situation can happen with alternating series where the terms are decreasing in magnitude and tend to zero, allowing the series to converge.

However, in the series \(\sum_{n=1}^{\infty} (-1)^{n+1} \cos^2(1/n)\), the terms \(\cos^2(1/n)\) do not tend to zero, as they approach 1, meaning they "stick" at 1. Due to this, the series fails the criteria for conditional convergence since the sequence of terms doesn't decrease to zero, which is crucial for alternating series to conditionally converge.

Understanding conditional convergence is vital for series analysis because it exhibits situations where a series appears to converge under specific arrangements, but not when absolute values are considered. This dichotomy underlines the delicate balance in convergence.

Alternating Series Test

The alternating series test, also known as the Leibniz test, evaluates whether an alternating series converges. An alternating series is of the form \(\sum_{n=1}^{\infty} (-1)^{n+1} a_n\), where the series has terms that alternately subtract and add. According to this test, for such a series to converge:

• The absolute values of the sequence \(a_n\) must be monotonically decreasing.
• The limit of the sequence as \(n\) approaches infinity must be zero: \(\lim_{n \to \infty} a_n = 0\).

In our original series, \(a_n = \cos^2(1/n)\), which do not decrease towards 0 as \(n\) becomes very large but rather approach 1. As a result, the series fails to meet the criteria for the alternating series test.

The alternating series test is a powerful tool, because if both conditions are satisfied, the series is guaranteed to converge. However, if these conditions aren’t met, as seen in our problem, the series does not converge. This test elegantly demonstrates the nuances of series convergence, highlighting the importance of term behavior in determining outcomes.