Examples

Maclaurin series of e to the x

What is the Maclaurin series of ?

Solution

Because , we find that for all .

So for all . Therefore for all by the Derivative-Coefficient identity.

Thus:

Maclaurin series of cos x

Find the Maclaurin series representation of .

Solution

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

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1
2
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5

By studying the generating pattern of the coefficients, we find for 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

Differentiate

Differentiate term-by-term:

Take negative because :

Final answer is

(b)

(1)

Recall the series

Compute the series for .

Set :

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(2) Compute the product.

Product of series:

(c)

(1)

Derivatives at are calculable from series coefficients.

Suppose we know the series

Then .

It may be easier to compute for a given than to compute the derivative functions and then evaluate them.

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(2) Compute .

Write the series such that it reveals the coefficients:

Coefficient with corresponds to the term with , not necessarily the term (e.g. if the first term is as here).

Compute :

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(3) Compute .

Use Derivative-Coefficient Identity:

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 that isn’t centered at .

The general format looks like this:

The coefficients satisfy . (Notice the .)

We find the coefficients by computing the derivatives and evaluating at :

By dividing by we can write out the first terms of the series: