# Worksheet: Taylor Series

In this worksheet, we will practice finding Taylor series of a function and finding the radius of convergence of the series.

**Q1: **

The following table shows the value of function and some of its derivatives at .

3 | 5 | 8 |

Write the first 5 terms of the Taylor series of .

- A
- B
- C
- D
- E

**Q2: **

For the function : , and for .

Find the first five terms of the Taylor series representation of at .

- A
- B
- C
- D
- E

**Q3: **

Consider the function .

Find the Taylor series representation of the function at .

- A
- B
- C
- D
- E

Are the terms of the Taylor series representation for the function finite or infinite?

- AInfinite
- BFinite

**Q4: **

Consider the function .

Find the Taylor series expansion of at .

- A
- B
- C
- D
- E

Write the Taylor series expansion of in sigma notation.

- A
- B
- C
- D
- E

**Q5: **

For a function : , and for . Find the Taylor series expansion of at .

- A
- B
- C
- D
- E

**Q6: **

The Taylor series representation of the function is given by .

Find the value of the fourth derivative of at .

- A
- B 0
- C 3
- D 6
- E 72

Find the value of the third derivative of at .

**Q7: **

What is the necessary and sufficient condition for the function to have a Taylor series representation about ?

- A is not equal to 0.
- B is not equal to for any positive integer . ( is the derivative of .)
- C exists for any positive integer . ( is the derivative of .)
- D exists.
- E is not equal to 0 for any positive integer . ( is the derivative of .)

**Q8: **

Consider the function .

Find the Taylor series representation of about .

- A
- B
- C
- D
- E

Find the interval of convergence of the Taylor series representation of about .

- A
- B
- C
- D
- E

What is the radius of convergence of the Taylor series representation of about ?

- A
- B1
- C
- D2
- E0

**Q9: **

What are the first four terms of the Taylor series of the function about ?

- A
- B
- C
- D
- E

**Q10: **

Consider the function .

Find the Taylor series representation of about .

- A
- B
- C
- D
- E

Find the interval of convergence of the Taylor series representation of about .

- A
- B
- C
- D
- E

What is the radius of convergence of the Taylor series representation of about ?

- A
- B
- C1
- D0
- E2