Leibniz series from the square wave

Find the Fourier series of the $2\pi$-periodic extension of

$$f(x)=\{\begin{array}{ll}-1& \text{if }-\pi \le x<0\\ 1& \text{if }0\le x<\pi \end{array}$$

and, by evaluating it at $x=\frac{\pi }{2}$, compute the sum of the numerical series $\sum _{k=0}^{\infty }\frac{(-1{)}^k}{2k+1}$.