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Does $(a_n) \to 0$ imply $\sum_{n=1}^{n=\infty} \frac {a_n}{n}$ is a convergent series?

It is simply a conjecture of mine that I couldn't disprove(or prove). I apologize if it has a simple counter-example.

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    $\begingroup$ Counterexample: $a_n = 1/\log n$ $\endgroup$ – user384138 Dec 10 '16 at 15:50

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