Fourier Series--Triangle Wave

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

$$b_n = {2\over L}\left\{{\int_0^{L/2} {x\over L/2}\sin\left({n\pi x\over... ...textstyle{1\over 2}}L)}\right]\sin\left({n\pi x\over L}\right)\,dx}\right\}\,dx$$

$$= {32\over\pi^2 n^2} \cos({\textstyle{1\over 4}}n\pi)\sin^3({\textstyle{1\over 4}}n\pi)$$

$$= {32\over\pi^2 n^2}\left\{\begin{array}{ll} 0 & \mbox{$n=0$, 4, \d... ... 6, \dots}\\ -{\textstyle{1\over 4}}& \mbox{$n=3$, 7, \dots}\end{array}\right.$$

$$= {8\over \pi^2 n^2}\left\{\begin{array}{ll} (-1)^{(n-1)/2} & \mbox{for $n$\ odd}\\ 0 & \mbox{for $n$\ even.}\end{array}\right.$$

The Fourier series is therefore

$$f(x)={8\over \pi^2}\sum_{n=1,3,5,\ldots}^\infty {(-1)^{(n-1)/2}\over n^2}\sin\left({n\pi x\over L}\right).$$

See also Fourier Series