This problem is designed to guide you through a “proof” of Plan-choral’s theorem, by start...

This problem is designed to guide you through a “proof” of Plan-choral’s theorem, by starting with the theory of ordinary Fourier series on a finite interval, and allowing that interval to expand to infinity.

(a) Dirichlet's theorem says that "any" function f(x) on the interval [–a, + a] can be expanded as a Fourier series:

Show that this can be written equivalently as

What is Cn, in terms of and bn ?

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(b) Show (by appropriate modification of Fourier's trick) that