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

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


Using the graph of 𝑦=1π‘₯ in the figure, we define π‘ŽοŠͺ to be the area that is shaded. This gives a term of the sequence π‘ŽοŠ.

Using an integral, give an exact expression for π‘ŽοŠ.

  • A(1+2+β‹―+𝑛)+(𝑛+1)ln
  • B(1+2+β‹―+𝑛)βˆ’(𝑛+1)ln
  • Cο€Ό1+12+β‹―+1π‘›οˆβˆ’(𝑛+1)ln
  • Dο€Ό1+12+β‹―+1π‘›οˆβˆ’(π‘›βˆ’1)ln
  • Eο€Ό1+12+β‹―+1π‘›οˆ+(𝑛+1)ln

The sequence π‘ŽοŠ is clearly increasing. What does rectangle 𝐴𝐡𝐢𝐷 tell us about the size of π‘ŽοŠͺ?

  • Aπ‘Ž<45οŠͺ
  • Bπ‘Ž>45οŠͺ
  • Cπ‘Ž=45οŠͺ
  • Dπ‘Žβ‰ˆ1οŠͺ

What, therefore, can you give as an upper bound on all the π‘ŽοŠ?

What can you conclude about the sequence π‘ŽοŠ?

  • AWe cannot conclude anything.
  • BIt is divergent.
  • CIts terms are eventually greater than 1.
  • DIt is convergent.
  • EIt converges to 1.


Let 𝑁(π‘₯)=π‘₯+32π‘₯.

Define 𝑁(π‘₯)=𝑁(π‘₯) rounded to 6 decimal places. Now let π‘₯=1, π‘₯=𝑁(π‘₯)=2.000000, π‘₯=𝑁(π‘₯)=1.750000, and so on. The sequence {π‘₯} is eventually constant. At what value is this?

With 𝑁(π‘₯)=𝑁(π‘₯) rounded to 10 decimal places, what is the limit, as π‘›β†’βˆž, of the sequence given by π‘₯=1 and π‘₯=𝑁(π‘₯) for 𝑛β‰₯1?

If π‘Žβ†’π‘§οŠ as π‘›β†’βˆž, then, by continuity of 𝑁, π‘Ž=𝑁(π‘Ž)→𝑁(𝑧). So 𝑁(𝑧)=𝑧. What would this 𝑧 be?

  • A√5
  • B√2
  • C√3


Using induction, show that the sequence 1,2,√7,3√7+1,ο„ž33√7+1+1,… is increasing and bounded, and find the limit of the sequence.

  • A3+√52
  • B3+√132


Consider the sequence (π‘Ž)∞ given by π‘Ž=2𝑛+35𝑛+6π‘›οŠοŠ©οŠͺ.

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 π‘Ž=8π‘₯βˆ’4π‘₯π‘₯+1.

  • AThe limit is 4.
  • BThe limit is 2.
  • CThere is no limit; the sequence tends to ∞.
  • DThere is no limit; the sequence tends to βˆ’βˆž.
  • EThe limit is βˆ’4.


The sequence π‘Ž=(2𝑛+1)βˆ’(2π‘›βˆ’1)loglog is convergent. What is its limit?


The completeness axiom can be used directly to show that a sequence is convergent. Consider the function 𝑁(π‘₯)=π‘₯2+1π‘₯ on nonzero numbers. Starting with π‘Ž=32, we can recursively define π‘Ž=𝑁(π‘Ž) for 𝑛β‰₯1.

What is π‘ŽοŠ«? Give your answer as a fraction.

  • A665,857470,832
  • B577408
  • C470,832665,857
  • D627,013,566,048886,731,088,897
  • E886,731,088,897627,013,566,048

By considering the decimal expansions of the sequence ο€Ήπ‘Žο…οŠ¨οŠ, it would seem that limοŠβ†’βˆžοŠπ‘Ž exists. What do you think this is?

  • A√2
  • B2

Suppose that π‘₯=π‘π‘ž with integers 𝑝, π‘ž. Write 𝑁(π‘₯) in its simplest form.

  • A2𝑝+2π‘žπ‘π‘žοŠ¨οŠ¨
  • B𝑝+2π‘žπ‘π‘žοŠ¨οŠ¨
  • C𝑝+π‘ž2π‘π‘žοŠ¨οŠ¨
  • D𝑝+2π‘ž2π‘π‘žοŠ¨οŠ¨
  • E2𝑝+π‘ž2π‘π‘žοŠ¨οŠ¨

What fact does the last result tell us about all the terms π‘ŽοŠ?

  • 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 𝑁(π‘₯)βˆ’2 in its simplest, factored form.

  • Aο€Ήπ‘₯βˆ’24π‘₯
  • Bο€Ήπ‘₯βˆ’1π‘₯
  • Cο€Ήπ‘₯+24π‘₯
  • Dο€Ήπ‘₯+2π‘₯
  • Eο€Ήπ‘₯βˆ’12π‘₯

What does the last result tell us about how all the π‘ŽοŠ and √2 are related?

  • Aπ‘Ž>√2
  • BIt tells us nothing.
  • Cπ‘Ž=√2 after some large 𝑛
  • Dπ‘Ž<√2

Write π‘₯βˆ’π‘(π‘₯) in its simplest form.

  • Aπ‘₯βˆ’22π‘₯
  • Bπ‘₯+2π‘₯
  • Cπ‘₯βˆ’12π‘₯
  • Dπ‘₯βˆ’1π‘₯
  • Eπ‘₯βˆ’2π‘₯

Fill in the blank: Given that ο€Ό32>2, the last two results imply that the sequence π‘ŽοŠ is .

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

The last result proves that π‘ŽοŠ really is convergent. Solving 𝑁(π‘₯)=π‘₯ says the limit is √2. Find a similar function for a recursive sequence convergent to √3.

  • A𝑁(π‘₯)=π‘₯3+1π‘₯
  • B𝑁(π‘₯)=π‘₯3βˆ’1π‘₯
  • C𝑁(π‘₯)=2π‘₯3βˆ’2π‘₯
  • D𝑁(π‘₯)=2π‘₯3+1π‘₯
  • E𝑁(π‘₯)=π‘₯3+2π‘₯

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