This is what I know so far:
- The given series uniformly converges by the M-test and that I can swap the integration and the sum when calculating the coefficients.
- Apparently I am supposed to use the trigonometric polynomial form of $f(x)$ to find the coefficients, but I am unsure as to how to go about proving that there is some Trigonometric polynomial that converges (uniformly?) to $f(x)$ (rather than just and approximation of it).
That said, please consider when you give an answer that this question was taken from an exam for first year math B.Sc. , and that I would love an answer that gives me pointers as to how to approach these kinds of questions.