TITLE: More convergent series
QUESTION [7 upvotes]: This question just reminded me of a conundrum I posed myself in my first year of university. I never did get a satisfactory answer...

Let $a_n$ be a null sequence. Does it follow that $\sum \frac{a_n}{n}$ converges?

Any ideas?

REPLY [13 votes]: If by null sequence you mean a sequence that converges to 0, then no. Try $a_n=1/\log n.$ By integral comparison, the series diverges:
$$\sum_2^\infty\dfrac1{n\log n}\geq\int_2^\infty\dfrac{dx}{x\log x}=\int_{\log 2}^\infty\dfrac{du}u=\infty,$$
where I've used the change of variables $u=\log x$.