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 = 1 𝑥 in the figure, we define 𝑎 4 to be the area that is shaded. This gives a term of the sequence 𝑎 𝑛 .

Using an integral, give an exact expression for 𝑎 𝑛 .

  • A ( 1 + 2 + + 𝑛 ) ( 𝑛 + 1 ) l n
  • B 1 + 1 2 + + 1 𝑛 + ( 𝑛 + 1 ) l n
  • C ( 1 + 2 + + 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 𝑛 is clearly increasing. What does rectangle 𝐴 𝐵 𝐶 𝐷 tell us about the size of 𝑎 4 ?

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

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

What can you conclude about the sequence 𝑎 𝑛 ?

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

Q2:

Let 𝑁 ( 𝑥 ) = 𝑥 + 3 2 𝑥 .

Define 𝑁 ( 𝑥 ) = 𝑁 ( 𝑥 ) rounded to 6 decimal places. Now let 𝑥 = 1 , 𝑥 = 𝑁 ( 𝑥 ) = 2 . 0 0 0 0 0 0 , 𝑥 = 𝑁 ( 𝑥 ) = 1 . 7 5 0 0 0 0 , 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 𝑛 1 ?

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

  • A 2
  • B 5
  • C 3

Q3:

Using induction, show that the sequence 1 , 2 , 7 , 3 7 + 1 , 3 3 7 + 1 + 1 , is increasing and bounded, and find the limit of the sequence.

  • A 3 + 1 3 2
  • B 3 + 5 2

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