# 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 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 as their radius of convergence.

In the following picture, 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 :

Plugging in t=0 yields the information that .

Let's take the first derivative:

Plugging in t=0 yields the information that .

Let's take the second derivative:

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

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

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

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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