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It's not for nothing that at college, the course in Fourier Series & Boundary Value Problems was better known as Mystery Math. There was a single key insight that made the entire course suddenly clear as a bell, and if you were lucky, that insight came _before_ the final exam.

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so, uh, you gonna tell us the insight?

Maybe that it's "just" a change of basis, onto a basis made of orthogonal periodic functions.

Change of bases being done with orthogonal projection derived from defining scalar product of two functions via finite integral of product of two functions: f dot g = integral f(x)g(x)dx over some interval.

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