# 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.

## 1 Answer

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

• im unsure how this would help as ∑ n=2 to infinity seems conditionally convergent but it is divergent and we need ∑nan to conditionally converge – Joemans Mar 17 '16 at 1:21
• No, the series I wrote is divergent. You'll need to make a suitable modification that's suggested by the series you wrote in your question to make it work. – user296602 Mar 17 '16 at 1:23
• Honestly thank you so much, but that didnt help me at all. – Joemans Mar 17 '16 at 1:38
• The example you gave is conditionally convergent because it's alternating. My series is not alternating. Change that fact by modifying my series. – user296602 Mar 17 '16 at 3:03