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
converges.
- 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