Convergence of $\sum{a_kb_k}$ if $\sum{a_k}$ converges and $\sum{b_k}$ absolutely converges. 
Convergence of $\sum{a_kb_k}$ if $\sum{a_k}$ converges and $\sum{b_k}$ absolutely converges.

I tried to think that
Since $\sum |b_k|$ is bounded
I thought that $\sum a_k b_k$ $<$ $S\sum a_k$. Is it a correct inequality?
If both sequences are just conditionally converge,
then do $\sum{a_k b_k}$ converge?

Because $\sum_ka_k$ converges, we have $\lim_ka_k=0$ so that $|a_k|$ is bounded by some $M>0$. If you assume that $\sum_k|b_k|$ converges, then $\sum_k|b_k|$ is Cauchy. Then, because
$$
\left|\sum_{k=m}^na_kb_k\right|\leq \sum_{k=m}^n|a_k||b_k|\leq M \sum_{k=m}^n|b_k|,
$$
we infer that $\sum_k a_kb_k$ is also Cauchy and thus converges.
You need the absolute convergence of $\sum_k b_k$. André's answer provides a neat example.
