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# If ∑ |an| converges prove ∑ an2 converges

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.

1. S says:

We can also use Abel’s test.

• S says:

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

2. Tyler says:

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.

3. M. Furquan says:

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.