Question

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $$ \sum_{n=2}^{\infty} \frac{(-1)^{n}}{n \ln n} $$

Short answer

By applying the Alternating Series Test, it is determined that the series \(\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n \ln n}\) is convergent.

Step-by-step solution

Identify the form of the series

The sum \(\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n \ln n}\) given in the exercise indicates that it is an alternating series because of the \((-1)^n\) term. Thus, Alternating Series Test is suitable for this.

Check if \(a_{n+1} \le a_{n}\) for all n

For all \(n\), \(\frac{1}{n \ln n} \ge \frac{1}{(n+1) \ln (n+1)}\). So, \(a_{n+1} \le a_{n}\) is satisfied. This condition essentially checks if the terms in the series are decreasing.

Check if \(\lim_{n\to\infty}a_{n}=0\)

Next, verify if the limit of the individual terms of the series is zero as \(n\) approaches infinity. Here, \(a_{n}=\frac{1}{n \ln n}\). We can see that as \(n\) approaches infinity, the denominator becomes very large and hence the term approaches zero. Thus, the second condition of the Alternating Series Test is also satisfied.