Consider the convergent sequence with terms defined by

Let . Find the value of and the value of such that for all for each of the following values of :

- ,
- ,
- ,
- ,
- .

First, we know

since , so by (10.10) on page 380 of Apostol we know the limit is 0.

Then we have,

We reversed the inequality sign in the final step since since . Thus, if then for every we have . We compute for the given values of as follows:

- implies .
- implies .
- implies .
- implies .
- implies .