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:

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