= Solution
A <Fourier series> represents a periodic <function> as a constant plus <sine> and <cosine> waves whose frequencies are integer multiples of its fundamental frequency. For period $2L$, use the convention
$$
f(x)\sim\frac{a_0}{2}+\sum_{n=1}^{\infty}\left[a_n\cos\left(\frac{n\pi x}{L}\right)+b_n\sin\left(\frac{n\pi x}{L}\right)\right],
$$
where the <Fourier coefficients> are
$$
\boxed{\begin{aligned}
a_0&=\frac1L\int_{-L}^{L}f(x)\,dx,\\
a_n&=\frac1L\int_{-L}^{L}f(x)\cos\left(\frac{n\pi x}{L}\right)\,dx,\quad n\geq1,\\
b_n&=\frac1L\int_{-L}^{L}f(x)\sin\left(\frac{n\pi x}{L}\right)\,dx,\quad n\geq1.
\end{aligned}}
$$
The coefficient formulas follow by multiplying by a basis <sine> or <cosine> and using their <orthogonality> over a full period. A <Fourier series> need not equal an arbitrary <function> at every <point>. The <pointwise convergence of a piecewise smooth Fourier series> theorem says that for a piecewise continuously differentiable <function>, the <Fourier series> converges at each <point> to the average of the two one-sided limits, and hence to $f(x)$ wherever $f$ is continuous.
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