Show that the sum of a convergent and divergent series is divergent

I am completely stumped on this one, so I could really use a hand with it.

Show that if the series $\displaystyle \sum a_k$ converges and the series $\displaystyle \sum b_k$ diverges, then the series $\displaystyle \sum (a_k+b_k)$ diverges.

A rigorous proof is expected, and we are to prove this by contradiction.

I don't know how I could answer this. This is all I really know that could possibly help me answer it.

If $\displaystyle \sum_{k=0}^{\infty}a_k$ converges and $\displaystyle \sum_{k=0}^{\infty}b_k$ converges, then $\displaystyle \sum_{k=0}^{\infty}(a_k+b_k)$

Moreover, if $\displaystyle \sum_{k=0}^{\infty}a_k=L$ and $\displaystyle \sum_{k=0}^{\infty}b_k=M$, then $\displaystyle \sum_{k=0}^{\infty}(a_k+b_k)=L+M$

I could really use some help with this one. I wasn't able to see my Calculus professor concerning the matter due to an unexplained absence on his part.