Examples

Maclaurin series of e to the x

What is the Maclaurin series of ?

Solution

Using repeatedly, we see that for all .

So for all . Therefore for all by the Derivative-Coefficient Identity:

Maclaurin series of cos x

Find the Maclaurin series representation of .

Solution

Use the Derivative-Coefficient Identity to solve for the coefficients:

0
1
2
3
4
5

By studying this pattern, we find the series:

Maclaurin series from other Maclaurin series

(a) Find the Maclaurin series of using the Maclaurin series of .

(b) Find the Maclaurin series of using the Maclaurin series of .

(c) Using (b), find the value of .

Solution

(a)

Remember that . Let us differentiate the cosine series by terms:

Take negative to get:

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(b)

Set :

Multiply all terms by :

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(c)

For any series:

we have:

We can use this to compute . From the series formula:

we see that:

Power, NOT term number

The coefficient with corresponds to the term having , not necessarily the term of the series.

Therefore:

Computing a Taylor series

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

Solution

A Taylor series is just a Maclaurin series centered at a nonzero number.

General format of a Taylor series:

The coefficients satisfy .

Find the coefficients by computing the derivatives and evaluating at :

The first terms of the series: