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.

Question

I think this is true but I'm not sure.

Answer 1

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 ...