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.

We can also use Abel’s test.

We cannot, because we don’t know that a_n is monotonic.

I think we could also prove this by showing that lim n goes to infinity of (an)^2/|an| = 0, so by theorem 10.9, (an)^2 converges since |an| does.

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.