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The Alternating Series Test (AST) is a method used to assess the convergence of series whose terms alternate in sign. An alternating series generally takes the form \( \sum_{n=1}^{\infty} (-1)^n a_n \, \text{or} \sum_{n=1}^{\infty} (-1)^{n+1} a_n,\) where \(a_n \) is a sequence of positive numbers. To apply AST, you need to verify two essential conditions:

• The sequence \(a_n\) is monotonically decreasing, meaning each term is smaller than or equal to the previous term \(a_{n+1} \le a_n \).


• The limit of the sequence approaches zero as \(n\) approaches infinity, \(\lim_{n \to \infty} a_n = 0\).

For example, in the series \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\), the corresponding non-alternating series is \(b_k = \frac{1}{k \ln k}\). First, confirm \(b_k\) is decreasing for \(k \ge 2\), which is true due to the logarithmic factor in the denominator. Next, check whether \(\lim_{k \to \infty} b_k = 0\), which holds as the natural logarithm increases. This ensures the series converges by AST, crucial for establishing whether it's conditionally or absolutely convergent.

Absolute convergence occurs when the series of absolute values \(\sum_{n=1}^{\infty} |a_n|\) converges. This is a stronger form of convergence, as every term's magnitude is summed, regardless of sign. To check absolute convergence for \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\), consider \(\sum_{k=2}^{\infty} \frac{1}{k \ln k}\). Since it resembles the harmonic series, which diverges, this series is expected to diverge as well.

Use the integral test and compare with \(\int_2^{\infty} \frac{1}{x \ln x} \, dx\). This integral diverges, confirming \(\sum_{k=2}^{\infty} \frac{1}{k \ln k}\) diverges as well. Therefore, the original series does not have absolute convergence. If a series does not converge absolutely, we look next to see if it converges conditionally.

Conditional convergence describes a series that converges when terms alternate in sign but diverges when terms’ absolute values are summed. After determining the original series \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\) converges by the Alternating Series Test, and the absolute series \(\sum_{k=2}^{\infty} \frac{1}{k \ln k}\) diverges, it fits the criteria for conditional convergence.

Here’s why:

• **AST Approved**: The alternating nature and conditions met ensure \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\) itself converges.


• **Absolute Series Divergence**: Just observing the results of the absolute series’ non-convergence validates the original series is not absolutely convergent.

Hence, our series’ convergence relies on its alternating structure, defining it as conditionally convergent. It provides rich insights into the behavior of series and highlights the diverse paths series can take in terms of convergence.