= Solution
In the real <Fourier series> convention for <period> $2\pi$,
$$
\boxed{y(x)\sim\frac{a_0}{2}+\sum_{n=1}^{\infty}[a_n\cos(nx)+b_n\sin(nx)],}
$$
where the <Fourier coefficients> are
$$
a_n=\frac1\pi\int_0^{2\pi}y(x)\cos(nx)dx\quad(n\geq0),\qquad
b_n=\frac1\pi\int_0^{2\pi}y(x)\sin(nx)dx\quad(n\geq1).
$$
These formulas apply to integrable periodic <functions>. Writing equality at each point requires appropriate convergence hypotheses; the periodic solution sought below is <smooth>, so its series causes no such difficulty.
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