Taylor Series


Introduction

Before you start this module, you must know how to find the Taylor polynomials of a given function. You should also be familiar with the geometric series, the notion of a power series, and in particular the concept of the radius of convergence of a power series.


Many functions can be written as a power series. The archetypical example is provided by the geometric series:

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which is valid for -1<x<1.

If we write a function as a power series with center tex2html_wrap_inline153 , we call the power series the Taylor series of the function with center tex2html_wrap_inline153 . (When the center is tex2html_wrap_inline157 , the Taylor series is also often called the McLaurin series of the function.)


Why does one care?

You probably like polynomials. Think of power series as "generalized" polynomials. Since (almost) all functions you encounter have a Taylor series, all functions can be thought of as "generalized" polynomials!

For instance, you will see that power series are easy to differentiate and integrate. No more techniques of integration, if one is satisfied with writing an integral as a power series!

In finding integrals and solving differential equations, one often faces the problem that the solutions can't be "found", just because they do not have a name, i.e., they cannot be written down by combining the familiar function names and the familiar mathematical notation. The error function tex2html_wrap_inline159 and the functions describing the motion of a "simple" pendulum are important examples. Power series open the door to explore even functions like these!


Taylor series as limits of Taylor polynomials

As you increase the degree of the Taylor polynomial of a function, the approximation of the function by its Taylor polynomial becomes more and more accurate.

It is thus natural to expect that the function will coincide with the limit of its Taylor polynomials! So you should expect the Taylor series of a function to be found by the same formula as the Taylor polynomials of a function: Given a function f(x) and a center tex2html_wrap_inline153 , we expect

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Finding the Taylor series of a function is nothing new! There are two problems, though.

1. It happens quite often that the right-hand side converges only for certain values of x. This is where the notion of the radius of convergence of a power series will become useful.

2. There are rare occasions, where the right-hand side is convergent, but does not equal the function f(x). I will not address this problem here; consult your book for a theorem usually called Taylor's Theorem.


An example.

Let's compute the Taylor series for tex2html_wrap_inline171 with center tex2html_wrap_inline157 . All derivatives are of the form tex2html_wrap_inline175 , so at tex2html_wrap_inline157 they evaluate to 1. Thus the Taylor series has the form:

displaymath169

Using the ratio test you can convince yourself that this power series converges everywhere; in fact it is equal to tex2html_wrap_inline175 for all values of x.


Another example.

Let's consider the natural logarithm tex2html_wrap_inline183 , and find its Taylor series with center tex2html_wrap_inline185 . Note that tex2html_wrap_inline157 would be an unwise choice! Why?

Here are the derivatives of tex2html_wrap_inline189 :

tex2html_wrap_inline189 , so tex2html_wrap_inline193 .

tex2html_wrap_inline195 , so f'(1)=+1.

tex2html_wrap_inline199 , so f''(1)=-1.

tex2html_wrap_inline203 , so f'''(1)=+2.

tex2html_wrap_inline207 , so tex2html_wrap_inline209 .

Do you see a pattern evolve? I bet you see that the next term will be tex2html_wrap_inline211 . It takes some practice to see how to translate this into a general formula:

We obtain that tex2html_wrap_inline213 (The factorial is 1 less than the order of the derivative.)

The alternating sign can be accommodated by inserting a term of the form tex2html_wrap_inline215 or tex2html_wrap_inline217 , depending on whether the first term (for n=1) is negative or positive!

Thus we see that tex2html_wrap_inline221 Check that this works for tex2html_wrap_inline223 . (It doesn't work for n=0; since this term is 0 anyway, we will omit it).

This yields the Taylor series

eqnarray100

In an earlier example (the example is almost identical!), we saw that this power series has a radius of convergence of 1. It turns out that the formula above is indeed valid for 0<x<2. (Recall that the center of the power series is 1.)


Try it yourself!

Find the Taylor series with center tex2html_wrap_inline157 for tex2html_wrap_inline231 .

Click here for the answer, or to continue.


Helmut Knaust
Sun Jul 14 23:59:04 MDT 1996

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