Worksheet: Monotone Convergence Theorem

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In this worksheet, we will practice using the monotone convergence theorem to test for convergence.

Q1:

Which of the following describes the least upper bound ℓ of a subset 𝑆 of numbers?

  • Athe number ℓ so that if 𝑏 ≥ 𝑠 for all 𝑠 ∈ 𝑆 , then ℓ ≤ 𝑏
  • Bthe number ℓ so that if 𝑏 ≤ 𝑠 for all 𝑠 ∈ 𝑆 , then ℓ ≥ 𝑏
  • Cthe largest number amongst all 𝑠 ∈ 𝑆
  • Dthe number ℓ so that if 𝑠 ∈ 𝑆 , then ℓ ≥ 𝑠
  • Ethe smallest number amongst all 𝑠 ∈ 𝑆

Q2:

Suppose 𝑆 is a subset of real numbers that has an upper bound 𝑏 .

Suppose that ℓ is a least upper bound of 𝑆 . Which of the following relations between ℓ and 𝑏 must be true?

  • A ℓ ≤ 𝑏
  • B ℓ ≠ 𝑏
  • C ℓ ≥ 𝑏
  • D ℓ < 𝑏
  • E ℓ = 𝑏

Suppose ℓ is a second least upper bound for 𝑆 . Which of the following relations between ℓ and ℓ must be true?

  • A ℓ > ℓ
  • B ℓ = ℓ
  • C ℓ ≠ ℓ
  • D ℓ < ℓ

Q3:

Consider the sequence 𝑎 = 3 𝑛 − 4 2 𝑛 − 1 for 𝑛 ≥ 1 .

Is 𝑎 > 𝑎 ?

  • Ano
  • Byes

Is 𝑎 > 𝑎 ?

  • Ano
  • Byes

Define 𝑃 , 𝑄 , 𝑅 , and 𝑆 by 𝑎 = 𝑃 𝑄 and 𝑎 = 𝑅 𝑆 . Write 𝑃 𝑆 − 𝑄 𝑅 in simplified form.

  • A 6 𝑛 − 1 0 𝑛 − 5
  • B 𝑛 + 3 𝑛 − 5
  • C 6 𝑛 − 1 0 𝑛 − 7
  • D 𝑛 − 3 𝑛 − 5
  • E 6 𝑛 + 1 0 𝑛 − 7

Using the above and the quadratic formula, find the smallest integer 𝑁 so that 𝑎 > 𝑎 whenever 𝑛 ≥ 𝑁 .

Q4:

Is the sequence 𝑎 = − 1 4 increasing, decreasing, or neither?

  • A 𝑎 is neither increasing nor decreasing.
  • B 𝑎 is decreasing.
  • C 𝑎 is increasing.

Q5:

The 𝑛 t h term of a sequence is 𝑎 . If 𝑎 = 𝑛 , is 𝑎 > 𝑎 ?

  • AYes
  • BNo

Q6:

Use < , = , or > to complete the following: A sequence with 𝑛 t h term 𝑎 is constant if 𝑎 𝑎 for each 𝑛 ≥ 1 .

  • A =
  • B >
  • C <

Q7:

Use < , = , or > to complete the following: A sequence with 𝑛 t h term 𝑎 is strictly increasing if 𝑎 𝑎 for each 𝑛 ≥ 1 .

  • A <
  • B >
  • C =

Q8:

Is the sequence 𝑎 = − 2 3 𝑛 + 1 9 increasing, decreasing, or neither?

  • A 𝑎 is increasing.
  • B 𝑎 is decreasing.
  • C 𝑎 is neither increasing nor decreasing.

Q9:

Is the sequence 𝑎 = 2 8 2 9 2 8 , where 𝑛 ≥ 1 increasing, decreasing, or neither?

  • A 𝑎 is decreasing.
  • B 𝑎 is increasing.
  • C 𝑎 is neither increasing nor decreasing.

Q10:

Is the sequence 𝑎 = 1 1 𝑛 + 4 4 increasing, decreasing, or neither?

  • Aincreasing
  • Bneither increasing nor decreasing
  • Cdecreasing

Q11:

Is the sequence 𝑎 = − 9 𝑛 − 3 2 increasing, decreasing, or neither?

  • Aincreasing
  • Bneither increasing nor decreasing
  • Cdecreasing

Q12:

Is the sequence 𝑎 = ( − 3 1 ) increasing, decreasing or neither?

  • Aincreasing
  • Bneither increasing nor decreasing
  • Cdecreasing

Q13:

Is the sequence 𝑎 = 1 1 9 𝑛 − 1 6 increasing, decreasing, or neither?

  • Aincreasing
  • Bneither increasing nor decreasing
  • Cdecreasing

Q14:

The 𝑛 t h term of a sequence is 𝑎 . If 𝑎 = 𝑛 , is 𝑎 > 𝑎 ?

  • ANo
  • BYes

Q15:

Is the sequence 𝑎 = − 1 1 + 2 𝑛 increasing, decreasing, or neither?

  • A 𝑎 is increasing.
  • B 𝑎 is decreasing.
  • C 𝑎 is neither increasing nor decreasing.

Q16:

Use < , = , or > to complete the following: A sequence with 𝑛 t h term 𝑎 is strictly decreasing if 𝑎 𝑎 for each 𝑛 ≥ 1 .

  • A <
  • B =
  • C >

Q17:

A geometric sequence has first term 𝑎 and common ratio 𝑟 . In which of the following cases will the sequence be decreasing?

  • A 𝑎 < 0 , − 1 < 𝑟 < 0
  • B 𝑎 > 0 , 0 < 𝑟 < 1
  • C 𝑎 > 1 , − 1 < 𝑟 < 0
  • D 𝑎 > 0 , − 1 < 𝑟 < 0
  • E 𝑎 < 0 , 0 < 𝑟 < 1

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