# $\sum a_{n}^{2}$ converges $\Rightarrow\sum \frac{a_{n}}{n}$ converges [duplicate]

$\sum a_{n}^{2}$ converges $\Rightarrow\sum \frac{a_{n}}{n}$ converges.

any hints?

## marked as duplicate by Thomas, Chris Janjigian, Ayman Hourieh, Daniel FischerJan 22 '15 at 19:31

detail: $$\left(\sum_{k=1}^n \frac{|a_k|}k \right)^2 \leq \sum_{k=1}^n \frac1{k^2} \sum_{k=1}^n |a_k|^2 \leq \sum_{k=1}^\infty \frac1{k^2} \sum_{k=1}^\infty |a_k|^2$$independant of $n$.
If these are sequences of real numbers, note that, for any $a,b\in \mathbb{R}$ we have $$0 \leq (a-b)^2,$$ and so $$ab \leq \frac{1}{2}(a^2 + b^2).$$
Can you pick $a$ and $b$ so that this gives $$\frac{a_n}{n} \leq \text{stuff}$$ where $\sum \text{stuff}$ is a convergent series?