Suppose that $\sum \limits_{n=1}^{\infty}a_n$ series with positive terms which diverges then series $\sum \limits_{n=1}^{\infty}\dfrac{a_n}{a_1+a_2+\dots+a_n}$ also diverges.
Can anyone show how to prove it? Unfortunately i have no ideas
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1One can sort of hack together a proof by "grouping" terms - i.e. group together all the terms where $2^n \leq a_1+a_2+\ldots+a_n < 2^{n+1}$, then bound the denominator as being at least $\frac{1}{2^{n+1}}$ for all these terms and show that the sum on top is "big enough" too. However, one needs to sort of hack together the case where a term transitions the sum from one group to the next, so I'm not sure this is a good approach. – Milo Brandt Jun 13 '16 at 04:02