Suppose $\displaystyle \sum_{k=1}^{\infty} (a_k+b_k)$ converges. Then $\displaystyle \sum_{k=1}^{\infty} a_k$ converges <===> $\displaystyle \sum_{k=1}^{\infty} b_k$ converges.

Should I prove this by contradiction using partial sums of the three series? (Headbang)