Fourier Series--Square Wave

Consider a square wave of length $2L$. Since the function is Odd, $a_0=a_n=0$, and

$$b_n = {2\over L}\int_0^L\sin\left({n\pi x\over L}\right)\,dx$$

$$= {4\over n\pi} \sin^2({\textstyle{1\over 2}}n\pi) = {4\over n\pi} ... ...begin{array}{ll} 0 & \mbox{$n$\ even}\\ 1 & \mbox{$n$\ odd.}\end{array}\right.$$

The Fourier series is therefore

$$f(x)={4\over\pi}\sum_{n=1,3,5,\ldots}^\infty {1\over n}\sin\left({n\pi x\over L}\right).$$

See also Fourier Series, Square Wave