$\sum a_n$ divergent with $\lim(n a_n)=0$

Question

Can we find an example of a divergent series $$\sum_{n=1}^\infty a_n$$ where the sequence $(a_n)$ is a decreasing sequence of real numbers, but such that $$\lim_{n\to\infty}(n a_n)=0$$

Answer 1

$$a_n = \frac{1}{n \log n}$$ for $n \geq 3$. By Cauchy Condensation, this grows like $$\frac{2^k}{2^k \cdot k \log 2} \sim \frac{1}{k}$$