# Property of summable sequences

Assume we have a non-negative sequence $\{b_n\}$ which is summable, i.e., $$\sum_{n=1}^\infty b_n <\infty.$$ Is it true that $b_n= O (1/n)$? My intuition tells me it is true (because that would mean that $b_n$ is dominated by the harmonic sequence, whose series diverges), but it also tells me to be careful with extremely strange sequences. I have not been able to formalize a proof neither to find such strange sequence.

Consider the sequence $$b_n=\cases{i/n & if n=10^i,i\in \Bbb N\\0& otherwise}$$ Then the series has finite sum $\frac{10}{89}$, but it is not $O(1/n)$.