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.