Worksheet: Monotone Convergence Theorem

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

  • Aℓ≥𝑏
  • Bℓ=𝑏
  • Cℓ<𝑏
  • Dℓ≤𝑏
  • Eℓ≠𝑏

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

  • Aℓ=ℓ
  • Bℓ<ℓ
  • Cℓ≠ℓ
  • Dℓ>ℓ

Q3:

Consider the sequence 𝑎=3𝑛−42𝑛−1 for 𝑛≥1.

Is 𝑎>𝑎?

  • Ano
  • Byes

Is 𝑎>𝑎?

  • Ano
  • Byes

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

  • A𝑛+3𝑛−5
  • B𝑛−3𝑛−5
  • C6𝑛+10𝑛−7
  • D6𝑛−10𝑛−7
  • E6𝑛−10𝑛−5

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

Q4:

Is the sequence 𝑎=−14 increasing, decreasing, or neither?

  • A𝑎 is neither increasing nor decreasing.
  • B𝑎 is decreasing.
  • C𝑎 is increasing.

Q5:

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

  • ANo
  • BYes

Q6:

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

  • A=
  • B<
  • C>

Q7:

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

  • A=
  • B<
  • C>

Q8:

Is the sequence 𝑎=−23𝑛+19 increasing, decreasing, or neither?

  • A𝑎 is decreasing.
  • B𝑎 is neither increasing nor decreasing.
  • C𝑎 is increasing.

Q9:

Is the sequence 𝑎=282928, where 𝑛≥1 increasing, decreasing, or neither?

  • A𝑎 is decreasing.
  • B𝑎 is neither increasing nor decreasing.
  • C𝑎 is increasing.

Q10:

Is the sequence 𝑎=11𝑛+44 increasing, decreasing, or neither?

  • Aneither increasing nor decreasing
  • Bincreasing
  • Cdecreasing

Q11:

Is the sequence 𝑎=−9𝑛−32 increasing, decreasing, or neither?

  • Aincreasing
  • Bdecreasing
  • Cneither increasing nor decreasing

Q12:

Is the sequence 𝑎=(−31) increasing, decreasing or neither?

  • Aneither increasing nor decreasing
  • Bdecreasing
  • Cincreasing

Q13:

Is the sequence 𝑎=119𝑛−16 increasing, decreasing, or neither?

  • Adecreasing
  • Bneither increasing nor decreasing
  • Cincreasing

Q14:

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

  • AYes
  • BNo

Q15:

Is the sequence 𝑎=−11+2𝑛 increasing, decreasing, or neither?

  • A𝑎 is increasing.
  • B𝑎 is neither increasing nor decreasing.
  • C𝑎 is decreasing.

Q16:

Use <, =, or > to complete the following: A sequence with 𝑛th term 𝑎 is strictly decreasing if 𝑎𝑎 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?

  • A𝑎<0, 0<𝑟<1
  • B𝑎<0, −1<𝑟<0
  • C𝑎>0, −1<𝑟<0
  • D𝑎>0, 0<𝑟<1
  • E𝑎>1, −1<𝑟<0

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