Using that
we obtain that
Both Taylor series expansions are valid for all values of x.
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 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.