# 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 of numbers?

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

Q2:

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

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

• A
• B
• C
• D
• E

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

• A
• B
• C
• D

Q3:

Consider the sequence for .

Is ?

• Ano
• Byes

Is ?

• Ano
• Byes

Define and by and . Write in simplified form.

• A
• B
• C
• D
• E

Using the above and the quadratic formula, find the smallest integer so that whenever .

Q4:

Is the sequence increasing, decreasing, or neither?

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

Q5:

The term of a sequence is . If , is ?

• ANo
• BYes

Q6:

Use , , or to complete the following: A sequence with term is constant if for each .

• A
• B
• C

Q7:

Use , , or to complete the following: A sequence with term is strictly increasing if for each .

• A
• B
• C

Q8:

Is the sequence increasing, decreasing, or neither?

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

Q9:

Is the sequence , where increasing, decreasing, or neither?

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

Q10:

Is the sequence increasing, decreasing, or neither?

• Aneither increasing nor decreasing
• Bincreasing
• Cdecreasing

Q11:

Is the sequence increasing, decreasing, or neither?

• Aincreasing
• Bdecreasing
• Cneither increasing nor decreasing

Q12:

Is the sequence increasing, decreasing or neither?

• Aneither increasing nor decreasing
• Bdecreasing
• Cincreasing

Q13:

Is the sequence increasing, decreasing, or neither?

• Bneither increasing nor decreasing
• Cincreasing

Q14:

The term of a sequence is . If , is ?

• AYes
• BNo

Q15:

Is the sequence increasing, decreasing, or neither?

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

Q16:

Use , , or to complete the following: A sequence with term is strictly decreasing if for each .

• A
• B
• C

Q17:

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

• A,
• B,
• C,
• D,
• E,