convergence divergence - Examples where series converges but product diverges and vice versa

Our professor gives us the following ungraded exercise for our analytic number theory class:

Let $E$ be a set with one element. Suppose $(b_n)$ is a sequence with $|b_n| \leq \lambda < 1$, and let $a_n = 1 + b_n$.

1) Find $(b_n)$ so that $\sum b_n$ converges, but $\prod a_n$ diverges.

2) Find $(b_n)$ so that $\prod a_n$ converges, but $\sum b_n$ diverges.

I am not sure how to do this problem. Any help is appreciated.