We find the Maclaurin series representation of a function.

### 1 Maclaurin Series

Maclaurin series are a special case of Taylor series with center . In this section we will develop the Maclaurin series for and and use these to create Maclaurin series of other, related functions.

**Maclaurin series**is a Taylor series with center 0. That is, a Maclaurin series is a power series representation for a function, , of the form

Recall the formula for the coefficients of a Taylor series centered at is . Substituting , we get the formula for the coefficients of a Maclaurin series: We now use this to create the Maclaurin series for .

Since for all whole numbers, , the coefficients are and the Maclaurin series representation is We can use the ratio test to verify that this representation converges for all and hence the representation is valid on the interval .

We now turn to two examples of finding Maclauirin series by making modifications to the previous example.

Replacing ‘’ in the Maclaurin series representation for with gives This representation is valid on the interval .

Replacing ‘’ with in the Maclaurin series representation for gives This representation is valid on the interval .

To find the Maclaurin series representation, we must determine the coefficients, . We will take advantage of a cyclical pattern in the derivatives of . The first four derivatives are: Since we arrived back at , the next four derivatives will be exactly the same. The coefficients require us to plug in the center, : Hence the numerators of the coefficients will cycle through the values and . Since the numerators of the even coefficients are 0, these coefficients are themselves 0 (because ). The numerators of the odd coefficients will alternate between and and hence

We can now write the Maclaurin series representation as This can be writtien in summation notation as The ratio test can be used to verify that this representation is valid on the interval .

We now consider an example of a Maclaurin series obtained by making modifications to the previous example.

We will use the Maclaurin series for with replacing and then multiply the result by . This gives This representation is valid in the interval .

Most of the work has already been done for us in the example. The numerators of the coefficients will cycle through the same four values as , but instead of starting with 0, they start with . Specfically, the numerators of the coefficients cycle through the values and . When the numerator is 0, the coefficient is 0, so the Maclaurin series representation for will only consist of even powers of : This can be writtien in summation notation as The ratio test can be used to verify that this representation is valid on the interval .

We now consider an example of a Maclaurin series obtained by making modifications to the previous example.

We will use the Maclaurin series for with replacing and then multiply the result by . This gives This representation is valid in the interval .