Given specific real Fourier coefficients, does there have to be a function that matches that?

Question

We were taught this in class:

Given the real numbers b1, b2,..., bn, there exists a cyclic function such that its non-zero Fourier coefficients are b1, b2, ... bn.

Can someone please explain why this is true?

Answer 1

One can just put formula for this function $$f(t) = \sum_{k=1}^{n} b_{k}e^{-k*t*i}$$