High School Math : Taylor and Maclaurin Series

Example Questions

Example Question #1 : Understanding Taylor Series

Give the   term of the Maclaurin series of the function

Explanation:

The   term of the Maclaurin series of a function  has coefficient

The second derivative of  can be found as follows:

The coeficient of  in the Maclaurin series is therefore

Example Question #2 : Understanding Taylor Series

Give the   term of the Taylor series expansion of the function  about .

Explanation:

The  term of a Taylor series expansion about  is

.

We can find  by differentiating twice in succession:

so the  term is

Example Question #1 : Taylor And Maclaurin Series

Give the  term of the Maclaurin series expansion of the function .

Explanation:

This can most easily be answered by recalling that the Maclaurin series for  is

Multiply by  to get:

The  term is therefore .

Example Question #2 : Understanding Maclaurin Series

Give the  term of the Maclaurin series of the function  .

Explanation:

The  term of a Maclaurin series expansion has coefficient

.

We can find  by differentiating three times in succession:

The term we want is therefore

Example Question #1 : Applying Taylor Series

Give the  term of the Maclaurin series expansion of the function .

Explanation:

The  term of a Maclaurin series expansion has coefficient

.

We can find  by differentiating twice in succession:

The coefficient we want is

,

so the corresponding term is .