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Possible Duplicate:
Contest problem about convergent series

Let ${p}_{n}\in \mathbb{R} $ be positive for every $n$ and $\sum_{n=1}^{∞}\cfrac{1}{{p}_{n}}$ converges,

How do I show that $\sum_{n=1}^{∞}{p}_{n}\cfrac{{n}^{2}}{{({p}_{1}+{p}_{2}+\dotso+{p}_{n})}^{2}}$ converges?

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    The problem is really interesting, since using the inequality proven in http://math.stackexchange.com/questions/214634/prove-that-sum-k-1n-frac2k1a-1a-2-a-k4-sum-k-1n-frac1a-k/223836#223836 it is possible to dramatically improve the bound given in the "official" answer relative to http://math.stackexchange.com/questions/200514/ - so, even if this is a duplicate question, I think it deserves an answer explaining the improvement.2012-11-21

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