# Give an example of a conditionally convergent series $\sum a_n$ such that $\sum n a_n$ is also conditionally convergent

As far as I can think of, the only conditionally convergent series are where $\sum na_n$ diverges. eg. $$\frac{(-1)^n}{n}$$ is conditionally convergent and $na_n = (-1)^n$ is divergent.

Hint: $$\sum_{n = 2}^{\infty} \frac{1}{n \log n}$$ is divergent by the integral test.