Fourier series are made up of sinusoids, all of which have frequencies that are integer multiples of some fundamental frequency. The trick, as with Taylor series, is to figure out what the coefficients are. In summation notation, we say (for odd functions of period 2, but that's just being picky in this context):
. . . and the trick is finding the coefficients ak.
You can find those coefficients by using calculus on complex exponentials, or you can use NuCalc and just build your function out of sines.
A great thing about using Fourier series on periodic functions is that the first few terms often are a pretty good approximation to the whole function, not just the region around a special point.
Fourier series are used extensively in engineering, especially for processing images and other signals. Finding the coefficients of a Fourier series is the same as doing a spectral analysis of a function.
Let's try to approximate a square wave (shown below) with a period of 2. One way to do it appears on the next page.
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