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The following is something that I think and hope is true. I tried searching for it online but with little success. A reference, proof, or counterexample would be nice.

Suppose $a_i $ is a positive sequence and $\sum_i a_i < \infty.$ Then there exists a positive sequence $b_i$ s.t $\sum_i b_i < \infty$ and $\sum_i \frac{a_i}{b_i} < \infty$.

Better2BLucky
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  • This is true, you can take $b_n=\sqrt {\sum_{i=n}^{\infty}a_i}$. See this question or this one for more information and some relevant links. – lulu Dec 01 '18 at 14:27
  • @lulu I saw this post - I don't think it's neceassily true that $\sum_i b_i < \infty$ if we do that – Better2BLucky Dec 01 '18 at 14:31
  • Yes....I don't see how to show that. Indeed, it looks like it might not be hard to make a counterexample to that. Odd. Rubin's claim regarding that series isn't very interesting if the sum diverges (not a proof of anything, clearly) – lulu Dec 01 '18 at 14:50

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