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.

Give examples to show that if $\displaystyle \sum a_k$ and $\displaystyle \sum b_k$ both diverge, then each of the series

$\displaystyle \sum (a_k + b_k)$ and $\displaystyle \sum (a_k - b_k)$

may converge or may diverge