If $\sum_{j=1}^\infty a_j$ is absolutely convergent, and $\sum_{j=1}^\infty b_j$ is conditionally convergent, can $\sum_{j=1}^\infty a_jb_j$ diverge? 
Suppose $\sum_{j=1}^\infty a_j$ is absolutely convergent, and $\sum_{j=1}^\infty b_j$ is only conditionally convergent. Is it possible for $\sum_{j=1}^\infty a_jb_j$ to diverge?

I am trying to come up with a counter example but I am having a hard time.
Any hints?

We must have $b_j \to 0$. Hence there is some $N$ such that $|b_j|<1$ $\forall j \ge N$. But then $|a_j b_j| \le |a_j|$ $\forall j \ge N$ . Then the series $\sum_{j=1}^\infty a_jb_j$ is  absolutely convergent.
