Extra convergent collection

This question simply advised me of a problem I postured myself in my first year of college. I never ever did get a sufficient solution ...

Let $a_n$ be a null sequence. Does it adhere to that $\sum \frac{a_n}{n}$ merges?

Any kind of suggestions?

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2019-05-04 18:38:35
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If by null series you suggest a series that merges to 0, after that no. Attempt $a_n=1/\log n.$ By indispensable contrast, the collection deviates :

$$\sum_2^\infty\dfrac1{n\log n}\geq\int_2^\infty\dfrac{dx}{x\log x}=\int_{\log 2}^\infty\dfrac{du}u=\infty,$$. where I've made use of the adjustment of variables $u=\log x$.

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2019-05-08 03:34:38
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