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Let $\sum_{n=1}^\infty{a_n}{}$ be a convergent series of positive terms.

Show that $$\sum_{n=1}^\infty\frac{\sqrt{a_n}}{n^p}$$

converges for p > $\frac{1}{2}$

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marked as duplicate by 23rd, Thomas Andrews, Davide Giraudo, Start wearing purple, Nick Peterson Aug 3 '13 at 12:59

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

You can use either AM-GM inequality or Cauchy-Schwartz inequality. – Sangchul Lee Aug 3 '13 at 12:24

Hint: by arithmetic-geometric mean, we have $$\frac {\sqrt{a_n}} {n^p} \leq a_n + \frac{1}{n^{2p}}.$$

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thanks a lot ;) – user85751 Aug 3 '13 at 12:31
You're very welcome! – Eric Auld Aug 3 '13 at 12:31
Now this is a very nice hint-answer. +1 – DonAntonio Aug 3 '13 at 12:45

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