If a_n does not equal zero for any n>=1 and ∑a_n converges absolutely, then ∑ 1/|a_n| diverges. The series are from n=1 to infinity.
I think this is true but I'm not sure.
If the series converges, then the terms must approach zero. In fact, all terms after the Nth (for some N) must be less than a, for some small a < 1.
So, since the terms are approaching zero, their reciprocals get larger and larger -- and there are infinitely many of them ...