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This question just reminded me of a conundrum I posed myself in my first year of university. I never did get a satisfactory answer...

Let $a_n$ be a null sequence. Does it follow that $\sum \frac{a_n}{n}$ converges?

Any ideas?

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    @Pete, I have no idea... was a long time ago- I just remembered the question out of the blue.2010-08-07

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If by null sequence you mean a sequence that converges to 0, then no. Try $a_n=1/\log n.$ By integral comparison, the series diverges:

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

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    thank$s$. I'm sure my younger self could have sworn \sum 1/nlog(n) converged. In fact I'm sure this was the motivating exam$p$le for me. As it happened, figured another counterexample as soon as I went to bed.2010-08-07