(12.4) Show for bounded sequences and . You may use the results of Exercise 9.9 from the book without first proving them. Hint: First show
and then apply Exercise 9.9(c).
Proof. Let and consider . As , . Similarly, . So,
so that is an upper bound for the set . In particular, this means that
Taking limits as of both sides preserves the inequality (this is the result of Exercise 9.9(c)), so that
and applying the definition of yields,
(12.12) This question is worth double points. Let be any sequence of non-negative real numbers. Let be defined by,
Hint: For the last inequality, show first that implies
Proof. We have trivially that , so we need only prove the first and last inequalities. First we will prove the inequality:
Consider and any . Consider any . We will bound from above:
where we have used that (1) as all are nonnegative, (2) , and (3) that for all we have . So,
As . This upper bound holds for all , so we may conclude that
As this relationship holds for all , taking limits of either side as (we may do this as increasing preserves the property that ) we obtain,
so that .
Taking limits then as we get that
We now show the first inequality, which uses similar techniques but is slightly different. We take , any , and consider arbitrary . Then we examine a lower bound for ;
where we have used that (1) as all are nonnegative, and (2) that for all we have . Notice now that , so that
This lower bound holds for all so we have proved that
Similar to before, taking limits as yields that
and taking limits as yields finally that
Show that if exists, then converges and
Proof. If exists then and so by (a) .
Give an example of a sequence so that converges but does not.
Here is one example:
Let . Then diverges, but what of ? For even we can see that and odd we have that . This means that and and we conclude that .
(14.1–4) For each of the following, determine whether the given series converges and justify your answer.
Converges by the ratio test. We have the ratios , whose limit is .
Converges by comparison, as whose corresponding series is a convergent geometric series.
Converges by the ratio test. We have the ratios
whose limit is .
(14.8) Show that if and are convergent series of nonnegative numbers, then converges. Hint: Show for all .
Proof. We claim that , from which convergence of the series follows from the comparison test.
Observe that as both and are nonnegative,
proving the claim.