= Solution
Use the real <Fourier series> convention
$$
f(x)\sim \frac{a_0}{2}+\sum_{n\ge1}\bigl(a_n\cos(2\pi nx)+b_n\sin(2\pi nx)\bigr).
$$
The constant coefficient is $a_0=2\int_0^1x(1-x)\,dx=1/3$. Reflection about $x=1/2$ changes the sine factor's sign and preserves the quadratic, giving $b_n=0$. Two <integrations by parts>, with $q=2\pi n$, give
$$
\int_0^1x(1-x)\cos(qx)\,dx=-\frac2{q^2},
\qquad a_n=-\frac1{\pi^2n^2}.
$$
Consequently the <Fourier series of a periodically extended quadratic arch> is
$$
\boxed{f_{\rm per}(x)=\frac16-\frac1{\pi^2}\sum_{n=1}^\infty\frac{\cos(2\pi nx)}{n^2}.}
$$
The periodic function is continuous, including at integers where its value is zero. Its piecewise smoothness gives pointwise <Fourier series> convergence everywhere, and the summable coefficient bound gives uniform and absolute convergence.
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