Worksheet: Convergent and Divergent Sequences

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

Q1:

Using the graph of 𝑦=1𝑥 in the figure, we define 𝑎 to be the area that is shaded. This gives a term of the sequence 𝑎.

Using an integral, give an exact expression for 𝑎.

  • A ( 1 + 2 + + 𝑛 ) + ( 𝑛 + 1 ) l n
  • B ( 1 + 2 + + 𝑛 ) ( 𝑛 + 1 ) l n
  • C 1 + 1 2 + + 1 𝑛 ( 𝑛 + 1 ) l n
  • D 1 + 1 2 + + 1 𝑛 ( 𝑛 1 ) l n
  • E 1 + 1 2 + + 1 𝑛 + ( 𝑛 + 1 ) l n

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

  • A 𝑎 < 4 5
  • B 𝑎 > 4 5
  • C 𝑎 = 4 5
  • D 𝑎 1

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

What can you conclude about the sequence 𝑎?

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

Q2:

Let 𝑁(𝑥)=𝑥+32𝑥.

Define 𝑁(𝑥)=𝑁(𝑥) rounded to 6 decimal places. Now let 𝑥=1, 𝑥=𝑁(𝑥)=2.000000, 𝑥=𝑁(𝑥)=1.750000, and so on. The sequence {𝑥} is eventually constant. At what value is this?

With 𝑁(𝑥)=𝑁(𝑥) rounded to 10 decimal places, what is the limit, as 𝑛, of the sequence given by 𝑥=1 and 𝑥=𝑁(𝑥) for 𝑛1?

If 𝑎𝑧 as 𝑛, then, by continuity of 𝑁, 𝑎=𝑁(𝑎)𝑁(𝑧). So 𝑁(𝑧)=𝑧. What would this 𝑧 be?

  • A 5
  • B 2
  • C 3

Q3:

Using induction, show that the sequence 1,2,7,37+1,337+1+1, is increasing and bounded, and find the limit of the sequence.

  • A 3 + 5 2
  • B 3 + 1 3 2

Q4:

Consider the sequence (𝑎) given by 𝑎=2𝑛+35𝑛+6𝑛.

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 𝑎=8𝑥4𝑥𝑥+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 .
  • EThe limit is 4.

Q6:

The sequence 𝑎=(2𝑛+1)(2𝑛1)loglog is convergent. What is its limit?

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