TITLE: $\sum a_n$ divergent with $\lim(n a_n)=0$
QUESTION [6 upvotes]: 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
$$\lim_{n\to\infty}(n a_n)=0$$

REPLY [10 votes]: $$a_n = \frac{1}{n \log n}$$ for $n \geq 3$. By Cauchy Condensation, this grows like $$\frac{2^k}{2^k \cdot k \log 2} \sim \frac{1}{k}$$