Find the Fourier Series for the Square Wave (Of Period $2 \pi$

Multiple Choice

Find the Fourier series for the square wave (of period $2 \pi$ ) given by $\mathrm{f}(\mathrm{x}) =\left\{\begin{array}{cc}-1, & -\pi \leq \mathrm{x}<0 \\ 1, & 0 \leq \mathrm{x} \leq \pi\end{array}\right.$

A) $\sum_{n=0}^{\infty} \frac{2}{\pi} \cdot \frac{\sin [(2 n+1) x]}{2 n+1}=\frac{2}{\pi}\left[\sin x+\frac{\sin 3 x}{3}+\frac{\sin 5 x}{5}+\frac{\sin 7 x}{7}+\ldots\right]$
B) $\sum_{n=0}^{\infty} \frac{4}{\pi} \cdot \frac{\sin [(2 n+1) x]}{2 n+1}=\frac{4}{\pi}\left[\sin x+\frac{\sin 3 x}{3}+\frac{\sin 5 x}{5}+\frac{\sin 7 x}{7}+\ldots\right]$
C) $\sum_{n=0}^{\infty} \frac{4}{\pi} \cdot \frac{\cos [(2 n+1) x]}{2 n+1}=\frac{4}{\pi}\left[\cos x+\frac{\cos 3 x}{3}+\frac{\cos 5 x}{5}+\frac{\cos 7 x}{7}+\ldots\right]$
D) $\sum_{n=0}^{\infty} \frac{2}{\pi} \cdot \frac{\cos [(2 n+1) x]}{2 n+1}=\frac{2}{\pi}\left[\cos x+\frac{\cos 3 x}{3}+\frac{\cos 5 x}{5}+\frac{\cos 7 x}{7}+\ldots\right]$