Taylor Series

Using that


we obtain that


Both Taylor series expansions are valid for all values of x.

Two Applications

The Error function

The integral of the function tex2html_wrap_inline99 can not be computed using the classical techniques of integration. Taylor series methods allow to write down this indefinite integral: Since


its indefinite integral, called the error function, has the Taylor expansion


Both series have tex2html_wrap_inline101 as their radius of convergence.

In the following picture, tex2html_wrap_inline99 is depicted in blue, while its integral is shown in red. The constant of integration has been chosen so that the red graph reflects the area under the blue graph.

The simple pendulum

The angular motion of a typical undamped simple pendulum can be described by the differential equation


Here x(t) denotes the angle (in radian measure) at time t between the pendulum and the resting position of the pendulum.

Let's suppose the pendulum starts at time t=0 in its resting position ,i.e., x(0)=0, with a certain initial angular velocity, say x'(0)=A. We want to describe the angular motion x(t) of the pendulum over time.

Suppose also that x(t) has a Taylor series with center tex2html_wrap_inline131 :


Plugging in t=0 yields the information that tex2html_wrap_inline135 .

Let's take the first derivative:


Plugging in t=0 yields the information that tex2html_wrap_inline139 .

Let's take the second derivative:


Now the differential equation tells us that tex2html_wrap_inline141 . Plugging in t=0 yields the information that tex2html_wrap_inline145 .

One more time. Let's take the third derivative:


Now the differential equation tells us that tex2html_wrap_inline147 by the chain rule. Plugging in t=0 yields the information that tex2html_wrap_inline151 , so tex2html_wrap_inline153 .

Thus the Taylor series for the pendulum motion starts out with


Actually computing a few more terms yields


Here are the graphs of the pendulum movements for various values of the initial velocity A.

Try it yourself!

For each function in the picture, move your arm to show the way the pendulum is swinging! What is the significance of the two thin red lines?

Helmut Knaust
Tue Jul 16 16:53:21 MDT 1996
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