Give example of a series $\sum a_n$ such that the series is conditionally convergent. and $\sum na_n$ is convergent

Question

I tried all the conditionally convergent series I know, I found $\sum na_n$ to be diverging for all of them.

But I am sure the question is correct

Answer 1

Let's try something like $a_n = \dfrac{(-1)^n}{n \ln n}$ for $n \ge 3$.

Then, $\displaystyle\sum_{n = 3}^{\infty}a_n = \sum_{n = 3}^{\infty}\dfrac{(-1)^n}{n \ln n}$ and $\displaystyle\sum_{n = 3}^{\infty}na_n = \sum_{n = 3}^{\infty}\dfrac{(-1)^n}{\ln n}$ converge by the alternating series test.

However, $\displaystyle\sum_{n = 3}^{\infty}|a_n| = \sum_{n = 3}^{\infty}\dfrac{1}{n \ln n}$ diverges by the integral test, so $\displaystyle\sum_{n = 3}^{\infty}a_n$ is conditionally convergent.