# Worksheet: Convergent and Divergent Sequences

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
• C
• D

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.

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?

• A
• B
• C

Q3:

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

• A
• B

Q4:

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.

Q5:

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 .

Q6:

The sequence is convergent. What is its limit?