Occasionally when teaching the sequences and series material in second-semester calculus I’ve included the following question as a bonus:
Question: Suppose is absolutely convergent. Does that imply anything about the convergence of ?
The answer is that converges. I’m going to give two proofs. The first is the more straightforward approach, but it is somewhat longer. The second is clever and shorter.
First method: If is absolutely convergent, then, by the divergence test, . Thus there exists some such that if then . This means that, for , . By the direct comparison test, then, converges.
Second method: As we argued above, . Thus . By the limit comparison test, then, converges.
(The first method is David Mitra’s answer and the second method is my answer to this Math.SE question.)