Consider the $2-$periodic function $f(x)$ defined $on-,by$ the formula $f(x)=cos(\frac{x}{2}).$

$(a)$ Find the Fourier series expansion $off(x).$

$(b)$ Use part $(a)to$ find the sum $of$ the following series:

$A={\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^{n+1}}{4n^2-1},$ and $,B={\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{4n^2-1}$