Consider the function \(f(x)=x,-\pi

a. Show that \(x=2 \sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sin n x}{n}\).

b. Integrate the series in part a and show that \(x^{2}=\frac{\pi^{2}}{3}-4 \sum_{n=1}^{\infty}\) \((-1)^{n+1} \frac{\cos n x}{n^{2}}\).

c. Find the Fourier cosine series of \(f(x)=x^{2}\) on \((0, \pi)\) and compare the result in part b.