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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.

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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 $O(1/n)$.

If the sequence was monotonous, however, then yes, it would be true.

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