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Let $a_n > 0$ and for all $n$ let $\sum\limits_{j=n}^{2n} a_j \le \dfrac 1n $ Prove or give a counterexample to the statement $\sum\limits_{j=1}^{\infty} a_j < \infty$

Not sure where to start, a push in the right direction would be great. Thanks!

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Think about what bounds you can put on these:

$b_k = \sum\limits_{j=2^k}^{2^{k+1}-1} a_j$

and note that $\sum\limits_1^\infty a_n = \sum\limits_0^\infty b_k$.

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    I think so, yes.2012-12-12
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Consider sum of sums $\sum_{i=1}^\infty \sum_{k=i}^{2k} a_j$.

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    Well, my formulation is not perfectly correct, but the idea should be evident.2012-12-12