Applications of Series
Now, that we know how to represent function as power series we can now talk about
at least a couple of applications of series.
There are in fact many applications of series, unfortunately most of them are beyond
the scope of this course. One application of power series (with the occasional use of
Taylor Series) is in the field of Ordinary Differential Equations when finding
Series
Solutions to Differential Equations
. If you are interested in seeing how that works
you can check out that chapter of my Differential Equations notes.
Another application of series arises in the study of Partial Differential Equations. One
of the more commonly used methods in that subject makes use of
Fourier Series
.
Many of the applications of series, especially those in the differential equations fields,
rely on the fact that functions can be represented as a series. In these applications it is
very difficult, if not impossible, to find the function itself. However, there are
methods of determining the series representation for the unknown function.
While the differential equations applications are beyond the scope of this course there
are some applications from a Calculus setting that we can look at.
Example 1
Determine a Taylor Series about
for the following integral.
Solution
To do this we will first need to find a Taylor Series about
for the integrand. This
however isn’t terribly difficult. We already have a Taylor Series for sine about
so
we’ll just use that as follows,
We can now do the problem.
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So, while we can’t integrate this function in terms of known functions we can come up with a
series representation for the integral.
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 Fall '08
 prellis
 Derivative, Power Series, Taylor Series, Partial differential equation

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