# Lesson Worksheet: Convergent and Divergent Sequences Mathematics • 12th Grade

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?

Q7:

The completeness axiom can be used directly to show that a sequence is convergent. Consider the function on nonzero numbers. Starting with , we can recursively define for .

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• B
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By considering the decimal expansions of the sequence , it would seem that exists. What do you think this is?

• A
• B2

Suppose that with integers , . Write in its simplest form.

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What fact does the last result tell us about all the terms ?

• ASome of them are integers.
• BThey are all rational numbers.
• CSome of them are irrational numbers.
• DThey are all integers.
• EThey are all irrational numbers.

Write in its simplest, factored form.

• A
• B
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• E

What does the last result tell us about how all the and are related?

• A
• BIt tells us nothing.
• C after some large
• D

Write in its simplest form.

• A
• B
• C
• D
• E

Fill in the blank: Given that , the last two results imply that the sequence is .

• AConvergent
• BMonotonically increasing
• CBounded
• DDivergent
• EMonotonically decreasing

The last result proves that really is convergent. Solving says the limit is . Find a similar function for a recursive sequence convergent to .

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• B
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• E