Let be a sequence of positive terms.
- Prove or give a counterexample: if diverges then also diverges.
- Prove or give a counterexample: if converges then also converges.
- Counterexample. Let . Then
diverges. Furthermore, and
- Proof. We know from a previous exercise (Section 10.20, Exercise #51) we know
for . So, for this case define . Then converges by hypothesis. Furthermore, (since so ),
Since this is the case , we have proved the statement