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?

Conditional convergence is not enough. Let $a_n=\dfrac{(-1)^n}{\sqrt{n}}$ and  $b_n=\dfrac{(-1)^{n+1}}{\sqrt{n}}$.
