Use of the above, along with Fourier's clever trick, to figure out a formula for any/all the coefficients of the Fourier series.
Example of a square wave, finding the coefficients, discussion of the "reasonableness" of the expansion that the formula yields.
(Showed Mathematica plots to see how nicely the think converges)
Discussion of conditions of Dirichlet, to decide when you can use a Fourier series. (Periodic, single-valued, finite number of maxima, minima, and discontinuities during one period, and the absolute value of the function should have a finite integral over one period)
Given the above, the series converges to the midpoint at a jump, otherwise, it converges to the function.
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