Infinite series question from analysis Let $a_n > 0$ and for all $n$ let $$\sum\limits_{j=n}^{2n} a_j \le \dfrac 1n $$  Prove or give a counterexample to the statement $$\sum\limits_{j=1}^{\infty} a_j < \infty$$
Not sure where to start, a push in the right direction would be great.  Thanks!

Consider sum of sums $\sum_{i=1}^\infty \sum_{k=i}^{2k} a_j$.
