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,

converges. However,

diverges.

Alternative proof:

hence, the convergence of implies that the sequence of partial sums of the series is bounded from above. By Theorem 10.7, the series converges.