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

any hints?


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

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use the Cauchy-Schwarz inequality.

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 $n$.

So the series is absolutely convergent.


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?


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