Solution:

The Fourier series of $f(x)$ with period $2 \pi$ is given by,

$$ f(x)=a_{0}+\sum_{n=1}^{\infty} a_{n} \cos n x+\sum_{n=1}^{\infty} b_{n} \sin n x \\ $$

$$ \begin{aligned} a_{0} &=\frac{1}{2 \pi} \int_{0}^{2 \pi} f(x) \mathrm{d} x \\\\ &=\frac{1}{2 \pi} \int_{0}^{2 \pi} x \mathrm{~d} x \\\\ &=\frac{1}{2 \pi}\left|\frac{x^{2}}{2}\right|_{0}^{2 \pi} \\\\ &=\frac{1}{2 \pi}\left(\frac{4 …