TITLE: Prove $\sum x_n < \infty \implies \sum \frac{\sqrt {x_n}}{n} < \infty$ if $x_n \geq 0$
QUESTION [5 upvotes]: Given $ \sum x_n  < \infty$ , I need to show that this implies $\sum \frac{\sqrt {x_n}}{n} < \infty $ if $x_n \geq 0$
I thought of using Abel's test to test the convergence but that test couldn't be applied here as $ \sum x_n  < \infty \nRightarrow  \sum \sqrt{x_n}  < \infty$  . I am completely stuck at this point. Any hint how to approach?

REPLY [6 votes]: Hint :
Compare your expression with : $$\sum \frac{1}{n^2} + \sum x_{n}$$
Hint 2: 
Cauchy inequality