If $\sum (a_n)^2$ converges and $\sum (b_n)^2$ converges, does $\sum (a_n+b_n)/n$ converge? Could someone help me to solve this or at least give me a hint? I've tried a few criterions and still can't really prove this, and I don't know what should I try. Any help would be appreciated

Applying Cauchy-Schwarz twice, you get
\begin{align*}
\Big(\sum_n \frac{a_n+b_n}n\Big)^2 &\leqslant \Big(\sum_n {a_n}^2+2\sum_{n}a_nb_n+\sum_n{b_n}^2\Big)\sum_n\frac 1{n^2}\\[5pt]
&\leqslant\Big(\sum_n {a_n}^2+2\sqrt{\sum_n{a_n}^2\sum_n{b_n}^2}+\sum_n{b_n}^2\Big)\sum_n\frac 1{n^2},
\end{align*}
where each sum clearly converges.
