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Lesson: Convergent and Divergent Sequences

Worksheet • 3 Questions

Q1:

Using the graph of 𝑦 = 1 π‘₯ in the figure, we define π‘Ž 4 to be the area that is shaded. This gives a term of the sequence π‘Ž 𝑛 .

Using an integral, give an exact expression for π‘Ž 𝑛 .

  • A ο€Ό 1 + 1 2 + β‹― + 1 𝑛  βˆ’ ( 𝑛 + 1 ) l n
  • B ο€Ό 1 + 1 2 + β‹― + 1 𝑛  βˆ’ ( 𝑛 βˆ’ 1 ) l n
  • C ο€Ό 1 + 1 2 + β‹― + 1 𝑛  + ( 𝑛 + 1 ) l n
  • D ( 1 + 2 + β‹― + 𝑛 ) + ( 𝑛 + 1 ) l n
  • E ( 1 + 2 + β‹― + 𝑛 ) βˆ’ ( 𝑛 + 1 ) l n

The sequence π‘Ž 𝑛 is clearly increasing. What does rectangle 𝐴 𝐡 𝐢 𝐷 tell us about the size of π‘Ž 4 ?

  • A π‘Ž < 4 5 4
  • B π‘Ž > 4 5 4
  • C π‘Ž β‰ˆ 1 4
  • D π‘Ž = 4 5 4

What, therefore, can you give as an upper bound on all the π‘Ž 𝑛 ?

What can you conclude about the sequence π‘Ž 𝑛 ?

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

Q2:

Let 𝑁 ( π‘₯ ) = π‘₯ + 3 2 π‘₯  .

Define 𝑁 ( π‘₯ ) = 𝑁 ( π‘₯ )  rounded to 6 decimal places. Now let π‘₯ = 1  , π‘₯ = 𝑁 ( π‘₯ ) = 2 . 0 0 0 0 0 0    , π‘₯ = 𝑁 ( π‘₯ ) = 1 . 7 5 0 0 0 0    , and so on. The sequence { π‘₯ }  is eventually constant. At what value is this?

  • A1.732051
  • B2.000000
  • C1.732143
  • D1.750000

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

  • A1.7320508076
  • B1.7320508097
  • C1.73214285714
  • D1.7320508078

If π‘Ž β†’ 𝑧  as 𝑛 β†’ ∞ , then, by continuity of 𝑁 , π‘Ž = 𝑁 ( π‘Ž ) β†’ 𝑁 ( 𝑧 )     . So 𝑁 ( 𝑧 ) = 𝑧 . What would this 𝑧 be?

  • A √ 2
  • B √ 3
  • C √ 5

Q3:

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

  • A 3 + √ 5 2
  • B 3 + √ 1 3 2
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