# [SOLVED] Infinite series proof.

• Nov 30th 2008, 11:17 AM
akolman
[SOLVED] Infinite series proof.
If $\sum ^\infty _{n=1} a_n$ converges and $a_n>0, \forall n \in \mathbb{N}$ , than does $\sum ^\infty _{n=1} \frac{\sqrt{a_n}}{n}$ converges??

I think it converges, but I don't know if I am right, or how to prove it.
• Nov 30th 2008, 01:38 PM
Opalg
Quote:

Originally Posted by akolman
If $\sum ^\infty _{n=1} a_n$ converges and $a_n>0, \forall n \in \mathbb{N}$ , than does $\sum ^\infty _{n=1} \frac{\sqrt{a_n}}{n}$ converges??

I think it converges, but I don't know if I am right, or how to prove it.

Geometric-arithmetic mean inequalty: $\left(a_nn^{-2}\right)^{1/2}\leqslant\tfrac12\left(a_n+n^{-2}\right)$.