# 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 .

358

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?

• AFinite
• BInfinite

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
• B3
• C72
• D0
• E6

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

• A2
• B
• C0
• D
• E1

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
• B2
• C
• D0
• E1

Q11:

Find the Taylor series of the function about .

• A
• B
• C
• D
• E

Q12:

Write the first four terms of the Taylor series expansion of in ascending powers of .

• A
• B
• C
• D
• E

Q13:

Write the first three terms of the Taylor series expansion of in ascending powers of .

• A
• B
• C
• D
• E

Q14:

Find the radius of convergence for the Taylor series of about .

• A
• B1
• C
• D
• E

Q15:

Find the radius of convergence for the Taylor series of about .

• A
• B
• C2
• D
• E