= Solution
Because the function is <even>, all sine <Fourier coefficients> vanish. Its average is zero, so $a_0=0$. The <Fourier cosine coefficients of a sine mode> follow from the product-to-sum identity:
$$
a_n=\frac2\pi\int_0^\pi\sin2x\cos nx\,dx=\begin{cases}\displaystyle\frac8{\pi(4-n^2)},&n\text{ odd},\\0,&n\text{ even}.\end{cases}
$$
For $n=2$, the integral is directly zero rather than an undefined substitution into a denominator. Thus
$$
\boxed{f(x)=\frac8\pi\sum_{m=0}^\infty\frac{\cos((2m+1)x)}{4-(2m+1)^2}.}
$$
The <Fourier series> equals the continuous function at every point, including the corners. Its coefficients decay quadratically, so the series is also <absolutely convergent>.
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