Using the graph of in the figure, we define to be the area that is shaded. This gives a term of the sequence .
Using an integral, give an exact expression for .
The sequence is clearly increasing. What does rectangle tell us about the size of ?
What, therefore, can you give as an upper bound on all the ?
What can you conclude about the sequence ?
Define rounded to 6 decimal places. Now let , , , and so on. The sequence is eventually constant. At what value is this?
With rounded to 10 decimal places, what is the limit, as , of the sequence given by and for ?
If as , then, by continuity of , . So . What would this be?
Using induction, show that the sequence is increasing and bounded, and find the limit of the sequence.