= Solution
With the usual real normalization, the <Fourier series> is
$$
f(x)\sim\frac{a_0}{2}+\sum_{n=1}^{\infty}\big[a_n\cos nx+b_n\sin nx\big],
$$
with <Fourier coefficients>
$$
\boxed{a_n=\frac1\pi\int_{-\pi}^{\pi}f(x)\cos nx\,dx\quad(n\geq0),\qquad
b_n=\frac1\pi\int_{-\pi}^{\pi}f(x)\sin nx\,dx\quad(n\geq1).}
$$
These formulas follow from <Fourier orthogonality>. Integrability is needed to define the coefficients, and periodicity alone does not guarantee pointwise equality with the series. For a piecewise smooth periodic <function>, the <Dirichlet-Jordan convergence theorem> gives the average of the one-sided limits at a jump and the <function> value at a <continuity> point.
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