Introduction

In this section we discussed a power series representation of a function. The problem with the approach in that section is that everything came down to needing to be able to relate the function in some way to:

taylor series

and while there are many functions out there that can be related to this function there are many more that simply can't be related to this.

So, without taking anything away from the process we looked at in the previous section, what we need to do is come up with a more general method for writing a power series representation for a function.

So, for the time being, let's make two assumptions. First, let's assume that the function f(x) does in fact have a power series representation about x = a,

taylor series

Next, we will need to assume that function, f(x), has derivatives of every order and that we can in fact find them all.

Now that we've assumed that a power series representation exists we need to determine what the coefficients, cn, are. This is easier than it might at first appear to be. Let's first just evaluate everything at x = a. This gives,

taylor series

So, all the terms except the first are zero and we now know what c0 is. Unfortunately, there isn't any other value of x that we can plug into the function that will allow us to quickly find any of the other coefficients. However, if we take the derivative of the function (and its power series) then plug in x=a we get,

taylor series

and we now know c1.

Let’s continue with this idea and find the second derivative.

taylor series

So, it looks like,

taylor series

Using the third derivative gives,

taylor series

Using the fourth derivative gives,

taylor series

Hopefully by this time you’ve seen the pattern here. It looks like, in general, we’ve got the following formula for the coefficients.

taylor series

This even works for n=0 if you recall that taylor series and define taylor series.

So, provided a power series representation for the function f(x) about x=a exists the Taylor Series for f(x) about x=a is,

Taylor Series

taylor series

If we use a=0, so we talking about the Taylor Series about x=0, we call the series a Maclaurin Series for f(x),

Maclaurin Series

taylor series

Before working any examples of Taylor Series we first need to address the assumption that a Taylor Series will in fact exist for a given function. Let’s start out with some notation and definitions that we’ll need.

To determine a condition that must be true in order for a Taylor series to exist for a function let’s first define the nth degree Taylor polynomial of f(x) as,

taylor series

Note that this really is a polynomial of degree at most n! If we were to write out the sum without the summation notation this would clearly be an nth degree polynomial. We’ll see a nice application of Taylor polynomials in the next section.

Notice as well that for the full Taylor Series,

taylor series

the nth degree Taylor polynomial is just the partial sum for the series.

Next, the remainder is defined to be,

taylor series

So, the remainder is really just the error between the function f(x) and the nth degree Taylor polynomial for a given n.

With this definition note that we can then write the function as,

taylor series

We now have the following Theorem.

References & Resources

  • http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx