Fourier Series of Even Square Wave

by sandy.bridge
Tags: fourier, series, square, wave
 P: 778 1. The problem statement, all variables and given/known data $-0.5\leq{t}\leq{1.5}, T=2$ The wave is the attached picture. I need to determine the Fourier Series of the wave in the picture. I know that $$f(t)=a_0+{\sum}_{n=1}^{\infty}a_ncos(n\omega_0t)+{\sum}_{n=1}^{\infty}b _nsin(n\omega_0t)$$ where $a_0=\bar{f}=0$ due to being an even function. Furthermore, $b_n=0$ due to being an even function also. That leaves, $$a_n=\int_{-0.5}^{0.5}cos(n\omega_0t)dt-\int_{0.5}^{1.5}cos(n{\omega}_0t)dt=0-\frac{1}{n\omega_0}(sin(1.5n{\omega}_0t)-sin(0.5n{\omega}_0t))=\frac{1}{n{\omega}_0}(sin(0.5n{\omega}_0t)-sin(1.5n{\omega}_0t))$$ therefore, $$f(t)=\frac{1}{{\omega}_0}\sum_{n=1}^{\infty}cos(n{\omega}_0t)(sin(0.5n{ \omega}_0t)-sin(1.5n{\omega}_0t))$$ Is this suffice as an answer, or am I missing something? My textbook is lacking examples so I just would like to know if I am doing it right. Thanks! Attached Thumbnails
 P: 778 I also had encountered another equation deeper in the chapter that states $$f(t)=A/2+(2A/\pi)\sum_{n=1}^{\infty}\frac{sin((2n-1)\omega{_0}t)}{2n-1}$$