# 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 for all , then
- Cthe largest number amongst all
- Dthe number so that if , then
- 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 ?

- AYes
- BNo

**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 increasing.
- B is decreasing.
- C is neither increasing nor decreasing.

**Q9: **

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

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

**Q10: **

Is the sequence increasing, decreasing, or neither?

- Aincreasing
- Bneither increasing nor decreasing
- Cdecreasing

**Q11: **

Is the sequence increasing, decreasing, or neither?

- Aincreasing
- Bneither increasing nor decreasing
- Cdecreasing

**Q12: **

Is the sequence increasing, decreasing or neither?

- Aincreasing
- Bneither increasing nor decreasing
- Cdecreasing

**Q13: **

Is the sequence increasing, decreasing, or neither?

- Aincreasing
- Bneither increasing nor decreasing
- Cdecreasing

**Q14: **

The term of a sequence is . If , is ?

- ANo
- BYes

**Q15: **

Is the sequence increasing, decreasing, or neither?

- A is increasing.
- B is decreasing.
- C is neither increasing nor 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 ,