Use the previous exercise (Section 10.4, Exercise #34) to establish each of the following limits.

- .
- .
- .
- .
- .
- .

- Let , then from Exercise #34 we know
where

Thus,

(since and then use the squeeze theorem). So,

- Let
So,

Thus,

- Let
Thus,

Therefore,

- Let
Thus,

So,

- Let
Thus,

Therefore,

- Let
Thus,

Therefore,

First, most of these functions are not increasing but this can be fixed by multiplying by -1, however there is big problem with (e) since it’s not monotonic… but I managed to prove it by substituting 2n instead of n and then breaking the sum into two parts one is usual since sin(xpi/2) is increasing on [0,1] and the other part you just change the indexing and you get a cosine which can be dealt with as usual.

part (f) is similar

How do you know you can do that substitution, and threat those two sequences as equivalent (at least when it comes to convergence)? Answers here suggest that it’s not true in general: https://math.stackexchange.com/q/4544928/942378.