So I encountered this problem in S. K. Mapa's book Introduction to Real Analysis. The answer to this question is given as below:
The series is $\sum_{n=2}^{\infty} \dfrac{(-1)^{n+1}}{n\space log\space n}$ convergent by Abel's test, since $\sum\dfrac{(-1)^{n+1}}{n}$ is a convergent series and the sequence $\{\frac{1}{log \space n}\}_{2}^{\infty}$ is a monotone decreasing sequence bounded below.
I don't understand how $\sum\dfrac{(-1)^{n+1}}{n}$ is a convergent series because $\sum\dfrac{1}{n}$ is a divergent series. If I take $\sum u_n = \sum \dfrac{(-1)^{n+1}}{n}$, then $\sum |u_n|= \sum \dfrac{1}{n}$ which is actually a divergent series and $u_n$ cannot be convergent.
But in my textbook, it says that series is convergent. How is it possible?
