Can we find a sequence $u\_n$ of positive real numbers such that 
$\sum\_{n=1}^\infty u\_n^2$ is finite, yet $\sum\_{n=1}^\infty ({u\_1+u\_2+...+u\_n\over n})^2$ is infinite ?


After several attempts, I think this is not possible, but I can't prove that the finiteness of the first sum implies the finiteness of the second sum.



Convergence of the sum of squares of averages of a sequence whose sum of squares is convergent
Hardy's inequality says
$$\sum\_{n=1}^{\infty}\left(\frac{a\_1+\cdots+a\_n}{n}\right)^p\le \left(\frac{p}{p-1}\right)^{p}\sum\_{n=1}^{\infty}a\_n^p$$
for any $p>1$.


