TITLE: Series problem! Can someone give me a counterexample?
QUESTION [9 upvotes]: Suppose positive series $\sum a_n<+\infty$, does this implies that
    $$\lim_{n\to\infty}na_n=0 .$$

REPLY [7 votes]: Let $a_n=\frac1n$ whenever $n$ is a positive perfect square, and $a_n=0$ otherwise. Then $\sum_{n\geq1}a_n=\sum_{m\geq1}a_{m^2}=\sum_{m\geq1}\frac1{m^2}=\frac{\pi^2}6<\infty$ but $\lim_{n\to\infty}na_n$ does not exist. If you don't like the terms $a_n=0$ in a positive series, replace them with something like $\exp(-n)$ that decreases sufficiently rapidly so as to not perturb the convergence of the series.