Maclaurin Series: Definition, Formula & Examples

In this video lesson, you will learn how the Maclaurin series is a special case of the Taylor series. You’ll also discover what some common Maclaurin series are for functions such as e^x and sin x.

Taylor Series

In math, when we get to the very complicated functions, we have other functions that help us approximate our more complicated functions, thus helping us solve them. One such approximation is called the Taylor series. The Taylor series of a particular function is an approximation of the function about a point (a) represented by a series expansion composed of the derivatives of the function. The formula for the Taylor series is this one:

maclaurin series

Looking at the expansion, we see that our first term is the function at the point a, the second term is the first derivative of the function at point a multiplied by (xa), then the third term is the second derivative of the function over a 2 factorial multiplied by (xa)2. We can see a pattern emerge. Each successive term is then found by following the pattern.

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We use this formula by finding our derivatives and then plugging those derivatives into the formula where it calls for them. Each of our derivatives is evaluated at the point a. We plug in our a value where the formula calls for it, too.

Maclaurin Series

A special case arises when we take the Taylor series at the point 0. When we do this, we get the Maclaurin series. The Maclaurin series is the Taylor series at the point 0. The formula for the Maclaurin series then is this:

maclaurin series

We use this formula in the same way as we do the Taylor series formula. We find the derivatives of the original function, and we use those derivatives in our series when it calls for it. The only difference is that we are now strictly using the point 0. All our derivatives are evaluated at the point 0.

Let’s look at a few examples of the Maclaurin series at work.

Ex

What is the Maclaurin series for the function f(x) = ex?

To find the Maclaurin series for this function, we first find the various derivatives of this function. This particular function is actually a very interesting function. All of its derivatives in fact are itself. So, the first derivative is ex, the second derivative is ex, and so on. Since we are looking at the Maclaurin series, we need to evaluate this function ex at the point 0. Since all the derivatives are the same, we evaluate ex at x = 0. We get e0 = 1. So all our derivatives will equal 1. Our Maclaurin series then becomes this:

maclaurin series

What we did was plug in 1 for all the derivatives since all our derivatives evaluated at the point 0 is equal to 1. The last two lines are our answer. The last line is the series written in summation form, and the line before that is the series expanded.

Sin x

Find the Maclaurin series for f(x) = sin x:

To find the Maclaurin series for this function, we start the same way. We find the various derivatives of this function and then evaluate them at the point 0. We get these for our derivatives:

Derivative At the point 0
f(x) = sin x f(0) = 0
f'(x) = cos x f'(0) = 1
f(x) = -sin x f(0) = 0
f'(x) = -cos x f'(0) = -1
f(x) = sin x f(0) = 0

We see our derivatives following a pattern of 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, etc. The numbers 0, 1, 0, and -1 keep repeating.

Plugging these into our Maclaurin series formula, we get this:

maclaurin series

Again, our last two lines are the answer with the last line being our answer written in summation form, and the line before that being our series expanded. Since we have the zeroes, when we write our answer, we skip over the zeroes.

Cos x

Find the Maclaurin series for f(x) = cos x:

What do we do first? We find the derivatives and evaluate at the point x = 0.

Derivative At the point 0
f(x) = cos x f(0) = 1
f'(x) = -sin x f'(0) = 0
f(x) = -cos x f(0) = -1
f'(x) = sin x f'(0) = 0
f(x) = cos x f(0) = 1

Just like our sine function has a pattern to the derivatives, our cosine function also has a pattern. We see that we have a repeating pattern of 1, 0, -1, and 0.

Plugging in our derivative values into our Maclaurin series formula, we get this:

maclaurin series

Since we have zeroes here, too, when we write our answer, we also skip over the zeroes. As you can see, our series skips every other term just like the sine function Maclaurin series.

1 /( 1 – X)

What is the Maclaurin series for the function f(x) = 1 / (1 – x):

First, we find the derivatives and then evaluate them at x = 0.

Derivative At the point 0
f(x) = 1 / (1 – x) f(0) = 1
f'(x) = (1 – x)-2 f'(0) = 1
f(x) = 2 (1 – x)-3 f(0) = 2
f'(x) = 6 (1 – x)-4 f'(0) = 6
f(x) = 24 (1 –x)-5 f(0) = 24

Plugging these into our formula, we get this:

maclaurin series

And we are done!

X3

All the functions we’ve seen so far has a series that keeps on going. There will be functions though that have a finite Maclaurin series. Look at this example.

Find the Maclaurin series for the function (x + 2)3:

First, we have to find the derivatives and evaluate at the point x = 0.

Derivative At the point 0
f(x) = (x + 2)3 f(0) = 8
f'(x) = 3 (x + 2)2 f'(0) = 12
f(x) = 6 (x + 2) f(0) = 12
f'(x) = 6 f'(0) = 6
f(x) = 0 f(0) = 0

Looking at this, we see that starting with the fourth derivative and thereafter, all the derivatives are 0. All our Maclaurin series terms from the fourth derivative onward will be 0. Our Maclaurin series then has a finite number of terms.

maclaurin series

Our Maclaurin series only has 4 terms. Again, it is because our derivatives evaluate to 0 after a certain point.

Lesson Summary

Let’s review what we’ve learned. The Taylor series of a particular function is an approximation of the function about a point (a) represented by a series expansion composed of the derivatives of the function. The formula for the Taylor series is this one:

maclaurin series

The Maclaurin series is the Taylor series at the point 0. The formula for the Maclaurin series then is this:

maclaurin series

To use these formulas, we find the derivatives and then evaluate them at the given point. Then we plug these derivatives into the formula. Sometimes, we will get an infinite series that we can write using summation notation. Other times, we get a finite series with just a few terms.

x

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