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Applications of Series

Applications of Series - Applications of Series Now that we...

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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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