In this worksheet, we will practice determining whether a sequence is convergent or divergent.

**Q1: **

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 .

- A
- B
- C
- D
- E

The sequence is clearly increasing. What does rectangle tell us about the size of ?

- A
- B
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- D

What, therefore, can you give as an upper bound on all the ?

What can you conclude about the sequence ?

- AIt is convergent.
- BWe cannot conclude anything.
- CIts terms are eventually greater than 1.
- DIt is divergent.
- EIt converges to 1.

**Q2: **

Let .

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?

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- C

**Q3: **

Using induction, show that the sequence is increasing and bounded, and find the limit of the sequence.

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- B