Let $|a_n|$ be a sequence such that $\lim_{n\rightarrow\infty} |a_n| = 0$. Will $\sum_{n=1}^\infty\frac{|a_n|}{n}$ converge? Its well understood that the harmonic series, $\sum_{n=1}^\infty\frac{1}{n}$ does not converge. One approach I tried to solve this problem was to show that there always exists some $0 < q \in \mathbb{R}$ and some $N \in\mathbb{N}$ such that for all $n > N$,
$$|a_n| < \frac{1}{n^q}$$
If this was true, then we know that
$$\sum_{n=N}^\infty \frac{|a_n|}{n} < \sum_{n=N}^\infty \frac{1}{n^{1 + q}} < \infty$$
but this seemed to go nowhere. The claim makes some intuitive sense to me, however I'm not sure if it is true or not. Is there a counter example?

Hint: What if $\displaystyle a_n=\frac1{\log(n+1)}$?
