Worksheet: Convergent and Divergent Sequences
In this worksheet, we will practice determining whether a sequence is convergent or divergent.
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 ?
- AWe cannot conclude anything.
- BIt is divergent.
- CIts terms are eventually greater than 1.
- DIt is convergent.
- EIt converges to 1.
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.
Consider the sequence given by .
State the first 5 terms of the sequence. If necessary, round your answers to 3 decimal places.
- A0, 0.455, 0.183, 0.124, 0.095
- B0.205, 0.17, 0.159, 0.153, 0.15
- C0.455, 0.318, 0.273, 0.25, 0.236
- D0.455, 0.183, 0.124, 0.095, 0.077
- E0, 0.205, 0.17, 0.159, 0.153
Find the limit of the sequence, if it exists.
Find the limit of the sequence whose terms are given by .
- AThe limit is 4.
- BThe limit is 2.
- CThere is no limit; the sequence tends to .
- DThere is no limit; the sequence tends to .
- EThe limit is .
The sequence is convergent. What is its limit?