- Prove that the series $\sum_{n=1}^\infty {|a_n b_n|}$ and $\sum_{n=1}^\infty {(a_n + b_n)^2}$ converges
- If the series $\sum_{n=1}^\infty {a^2_n}$ and $\sum_{n=1}^\infty {b^2_n}$ converges.
The part $\sum_{n=1}^\infty {(a_n + b_n)^2}$ does not necessarily converge, I had thought so, but I am doubting the first condition. Someone can help me, please.