Question

Determine whether the series converges conditionally or absolutely, or diverges. $$ \sum_{n=2}^{\infty} \frac{(-1)^{n}}{\ln n} $$

Short answer

The series \(\sum_{n=2}^{\infty} \frac{(-1)^{n}}{\ln n}\) converges conditionally.

Step-by-step solution

Identify the Series

The series is an alternating series because it includes \((-1)^n\) which alternates signs. The terms of the series are given by \(a_n = \frac{1}{\ln n}\).

Apply the Alternating Series Test

The first step in the Alternating Series Test is to show that \(a_{n+1} < a_n\), that is, the terms of the series are decreasing. Taking the derivative of \(a_n\) gives \(-\frac{1}{n(\ln n)^2}\) which is always less than 0 for \(n \geq 2\), proving that the terms are decreasing. The second step is to show that \(\lim_{n \rightarrow \infty} a_n = 0\). Taking the limit as \(n\) approaches infinity gives 0. Hence, by the Alternating Series Test, the series converges.

Apply the Absolute Convergence Test

Absolute convergence is tested by taking the absolute value of the series terms and determining if that series converges. The absolute value of the series is given by \(\sum_{n=2}^{\infty} \frac{1}{\ln n}\). This series does not converge, as it can be shown using the Integral Test that it behaves similarly to the harmonic series, which is known to diverge. Thus, the series does not converge absolutely.