Cálculo de Integrales de funciones expresadas como serie de Taylor

Cálculo de Integrales de funciones expresadas como serie de Taylor.

Applications of Taylor Series

We started studying Taylor Series because we said that polynomial functions are easy and that if we could find a way of representing complicated functions as series ("infinite polynomials") then maybe some properties of functions would be easy to study too. In this section, we'll show you a few ways in Taylor series can make life easy.

Evaluating definite integrals

Remember that we've said that some functions have no antiderivative which can be expressed in terms of familiar functions. This makes evaluating definite integrals of these functions difficult because the Fundamental Theorem of Calculus cannot be used. However, if we have a series representation of a function, we can oftentimes use that to evaluate a definite integral.

Here is an example. Suppose we want to evaluate the definite integral

The integrand has no antiderivative expressible in terms of familiar functions. However, we know how to find its Taylor series: we know that

Now if we substitute , we have

In spite of the fact that we cannot antidifferentiate the function, we can antidifferentiate the Taylor series:

Notice that this is an alternating series so we know that it converges. If we add up the first four terms, the pattern becomes clear: the series converges to 0.31026.

Understanding asymptotic behaviour

Sometimes, a Taylor series can tell us useful information about how a function behaves in an important part of its domain. Here is an example which will demonstrate.

A famous fact from electricity and magnetism says that a charge q generates an electric field whose strength is inversely proportional to the square of the distance from the charge. That is, at a distance r away from the charge, the electric field is

where k is some constant of proportionality.

Oftentimes an electric charge is accompanied by an equal and opposite charge nearby. Such an object is called an electric dipole. To describe this, we will put a charge q at the point and a charge -q at .

Along the x axis, the strength of the electric fields is the sum of the electric fields from each of the two charges. In particular,

If we are interested in the electric field far away from the dipole, we can

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