# Taylor Series: Definition, Formula & Examples

## What Is the Taylor Series?

World politics can be complicated. Distant economies change dramatically. Far away populations move their centers. Sometimes, it's better to focus on what is happening locally. The same may be said for complicated mathematical expressions. By looking at the localized behavior of a function, we can often gain amazing insights. This is where the Taylor series is really useful. Simply put, the **Taylor series** is a representation of a function that can help us do mathematics. Uses of the Taylor series include analytic derivations and approximations of functions.

If we have a function which we can differentiate, then we can express that function as a Taylor series with the formula you're looking at on your screen right now.

Here are a few comments about notation and other particulars for this series. First of all, the **derivative** could be written as the formula you're now looking at on your screen:

This is telling us to differentiate with respect to *x* and then to substitute the *x* variable with the *a* variable. We have not completely eliminated the *x* variable; it still appears in the Taylor series formulation, only the *x* in the derivative gets replaced by *a*. There is also a factorial notation, which is shown by the exclamation point. It means, n! = *n*(*n* - 1)(*n* - 2) and so on. For example, 3! = 3(2)(1) = 6.

Although the Taylor series has an infinite number of terms, we often keep only a few terms. How many terms we keep is determined by knowing the convergence of the series. Basically, **convergence** means that as we include more and more terms, the sum of terms is not growing without bounds. A series may not converge. Analyzing how and if a series converges will tell us the values of *x* over which this series is valid.

In the examples used in this lesson, we will give this region of convergence information and not have to determine it ourselves. The *a* term in this series formula allows us to offset the series. We can then focus the series at a particular value on the *x*-axis. For this lesson, we will refer to the *a* term as the offset term.

## Making Sense of the Offset Term

Let's determine the Taylor series for a particular function. Then we will plot a few terms to help us make sense of the *a* term. Consider the following function:

From the Taylor series formula we see that we need derivatives of *f*(*x*). To make this easier, we write the following:

Then, the first derivative is:

And the second derivative of *f*2(x) equals:

Our Taylor series for this function, about the point *a*, is thus the following:

for values of *x* between -1 and +1.

Let's say that we would like to match our curves at and near the point *a* = -0.5. In the following plot, the dark line is the desired function. The colored curve is the sum of only the first three terms of the Taylor series for *a* = -0.5. Note that *f*(*x*) is plotted as *y* in this and subsequent figures:

Even with only three terms, we see that the curves match very well near *x* = -0.5, but not so well at *x* = 0. If we want a better match at *x* = 0, we let *a* = 0. The effect is to offset the series. Now our graph becomes what you're looking at on screen now:

With this change, we have a much better match at *x* = 0. Thus, we can focus where we would like the best match to occur by selecting the value for *a* to be at or near that point. The *a* = 0 location is special. There is considerable interest in the Taylor series near the origin. Let's use our example to show why this is true. If we go back to our Taylor series for this example and let *a* = 0, we get:

This example is a much simpler looking expression. This is called a **MacLaurin series** and it's a Taylor series evaluated at *a* = 0. It usually yields a much simpler expression. This simplicity is often advantageous for analytic work. For example, we're now well equipped to show that the derivative of the sine is the cosine.

## Taylor Series Verifying Derivatives

The derivative of the sine function gives us the cosine function. This identity can be proven by first writing the Taylor series for sin(*x*). We will let *a* = 0 to simplify the analysis. Then we will differentiate this series term by term. The resulting series will be the Taylor series for cos(*x*). For *f*(*x*) = sin(*x*), we can prepare our work by listing some derivatives like the ones on your screen right now:

Substituting this information into our Taylor series formula gives us the formula that you're now looking at on your screen, which converges for all values of *x*:

Now, we let *a* = 0, noting that sin(0) = 0 and cos(0) = 1, which you can now see reflected in the equations on your screen.

To verify our trig derivative identity, let's say that we have already worked out or are given that the Taylor series for cos(*x*) can be what you're now looking at on your screen, which is for all values of *x*.

Now, differentiate each term of the sin(*x*) Taylor series and observe the results:

We have used the Taylor series to verify that the derivative of sin(*x*) is cos(*x*).

## Lesson Summary

Let's review what we've learned. The **Taylor series** is a mathematical series expression for differential functions. The complete description of the series expression includes the region of convergence. We can evaluate the series about a value of *x* by selecting an offset term *a*. The Taylor series can sometimes be called a **MacLaurin series**, which is a Taylor series evaluated at *a* = 0. In this lesson, we have used the Taylor series to approximate a function and to analytically verify a trig derivative identity.

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## Taylor Series - Applications

### Review Topics

The **Taylor series** of a differentiable function, **f(x)** about an offset term **a** is given by the following series

A Taylor series is **convergent** if the sum of infinitely many terms is a finite number. To determine the convergence of a series, we usually apply a convergence test, like the ratio test.

### Taylor Series - Applications

Taylor series are used to approximate complex functions with polynomial functions, which are much easier to differentiate and integrate. The only disadvantage is that the function may be approximated with the Taylor series on a restrictive domain, the interval of convergence of the series.

For example, the exponential function is represented by the Taylor series around zero for any **x**, because the interval of convergence is the entire real axis,

To see how Taylor series is used to do algebra easier, we will evaluate the integral

### Application

1) Using the Taylor series of the exponential function, given above, write the Taylor series of

2) Integrate the first three terms and the general term of the Taylor series obtained in 1).

3) Write the Taylor series around zero of the given integral.

### Solution

1)

2. An antiderivative of the first three terms is

An antiderivative of the general term is

3. The general antiderivative of the given exponential function is

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