Answer
The series $\sum_{n=2}^{\infty} \frac{(-1)^{n}}{ \ln n}$ converges.
Work Step by Step
The positive, decreasing term is $b_n=\frac{1}{\ln n}$; now we have
$$\lim_{n\to \infty } b_n=\lim_{n\to \infty } \frac{1}{\ln n}=\frac{1}{\infty}=0.$$
Hence, using the alternating series test, the series $\sum_{n=2}^{\infty} \frac{(-1)^{n}}{ \ln n}$ converges.
