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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, O.L., Nicholas R. Peterson Aug 3 '13 at 12:59

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You can use either AM-GM inequality or Cauchy-Schwartz inequality. –  sos440 Aug 3 '13 at 12:24

1 Answer 1

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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