How can I show this inequality between series

Asked Sep 03 '16 at 17:39

Active Sep 03 '16 at 19:44

Viewed 46 times

Prove that $\forall \{A_n\}_{n \in \mathbb{N}}$ $\exists k\in \mathbb{R}$ such that $\displaystyle \sum_{n=1}^{\infty}{\dfrac{n}{A_1+\cdots+A_n}}\leq k \cdot\sum_{n=1}^{\infty}{\dfrac{1}{A_n}}$

I already know that the case when $\displaystyle \sum_{n=1}^{\infty}{A_n}$ converges we gey the inequality $\infty\leq\infty$ but I don't know what to do when $\sum{A_n}$ diverges.

Edit: we are assuming that $\displaystyle \sum_{i=1}^{j}{A_i}\neq 0 \quad \forall j\in\mathbb{N}$

edited Sep 03 '16 at 19:44

asked Sep 03 '16 at 17:39

Nah

Presumably $A_i$ is also non-negative? – Alex R. Sep 03 '16 at 18:49
i think, but the enunciate doesn't say anything – Nah Sep 03 '16 at 18:51
Do we assume the series on the LHS is converging? – iamvegan Sep 03 '16 at 19:39
Its converges for $\sum{A_n}$ diverging sufficiently fast (>n^2) but I'm not sure what happens in the other case – Nah Sep 03 '16 at 19:45
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I think this problem has been discussed here. – iamvegan Sep 03 '16 at 20:55

How can I show this inequality between series

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