Prove that if the series converges then the series also converges. Also, give an example to show that the converse is false, i.e., a case in which converges but does not.
Proof. Assume converges. Then we know . Thus, by the definition of the limit, for all there exists an integer such that
Taking we then know there exists an such that
But then, if . Thus, we have
The first sum is a finite sum so it converges, and the second sum converges by comparison with . Hence, converges
Counterexample. The converse is false. Let . Then,