My intuition says yes, but I'm trying to prove it using limit laws and getting stuck. So far I have the following:
Let $(s_n)$ be the sequence of partial sums of $\sum(a_n+b_n)$ and $(t_n)$ be the sequence of partial sums of $\sum(a_n-b_n)$. So $\lim s_n=\sum(a_n+b_n)$ and $\lim s_n=\sum(a_n-b_n)$. Then $\lim(s_n+t_n)=\lim s_n+\lim t_n$ (because $(s_n),(t_n)$ converge), and then $\lim(s_n+t_n)=\lim((a_n+b_n)+(a_n-b_n))=\lim(2a_n)=\sum(a_n+b_n)+\sum(a_n-b_n)$.
But I'm not convinced that actually goes anywhere, or if I'm taking the right approach. If anyone has any ideas to help in making this proof work out I would really appreciate it.