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 of a subset 𝑆 of numbers?

  • Athe number so that if 𝑏𝑠 for all 𝑠𝑆, then 𝑏
  • Bthe number so that if 𝑠𝑆, then 𝑠
  • Cthe number so that if 𝑏𝑠 for all 𝑠𝑆, then 𝑏
  • Dthe largest number amongst all 𝑠𝑆
  • Ethe smallest number amongst all 𝑠𝑆

Q2:

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

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

  • A𝑏
  • B=𝑏
  • C<𝑏
  • D𝑏
  • E𝑏

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

  • A=
  • B<
  • C
  • D>

Q3:

Consider the sequence 𝑎=3𝑛42𝑛1 for 𝑛1.

Is 𝑎>𝑎?

  • Ano
  • Byes

Is 𝑎>𝑎?

  • Ano
  • Byes

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

  • A𝑛+3𝑛5
  • B𝑛3𝑛5
  • C6𝑛+10𝑛7
  • D6𝑛10𝑛7
  • E6𝑛10𝑛5

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

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 𝑛1.

  • A=
  • B<
  • C>

Q7:

Use <, =, or > to complete the following: A sequence with 𝑛th term 𝑎 is strictly increasing if 𝑎𝑎 for each 𝑛1.

  • 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 𝑛1 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=9𝑛32 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=119𝑛16 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 𝑛1.

  • 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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