If $\sum (a_n)^2$ converges and $\sum (b_n)^2$ converges, does $\sum (a_n+b_n)/n$ converge

Question

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

Answer 1

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.