Does $\sum _{n=1}^{\infty }\left(a_n\:+\:b_n\right)^2$ converges?

Question

I'm trying to understand if the following holds true

If $\sum _{n=1}^{\infty }\:a{^2}_n$ and $\sum _{n=1}^{\infty }\:b{^2}_n$ are two convergent series, does $\sum _{n=1}^{\infty }\left(a_n\:+\:b_n\right)^2$ converges?

The series is non-negative and I believe I should use the convergent comparison test in order to determine if it converges, but I feel a bit clueless here.

Answer 1

We have $\left(\displaystyle\sum|a_{n}||b_{n}|\right)^{2}\leq\displaystyle\left(\sum|a_{n}|^{2}\right)\left(\sum|b_{n}|^{2}\right)$ and then $|a_{n}+b_{n}|^{2}\leq|a_{n}|^{2}+2|a_{n}||b_{n}|+|b_{n}|^{2}$, so the result follows.