## Calculus (3rd Edition)

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