Question Number 145981 by Gbenga last updated on 10/Jul/21
![let f(x) be a function period 2π such that:f(x)={x, 0<x<π {π, π<x<2π show that the fourier series for f(x) in the interval 0<x<2π is ((3π)/4)−(2/π)[cosx+(1/3^2 )cos3x+(1/5^2 )cos5x+...]−[sinx+(1/2)sin2x+(1/3)sin3x+...]](https://www.tinkutara.com/question/Q145981.png)
$${let}\:{f}\left({x}\right)\:{be}\:{a}\:{function}\:{period}\:\mathrm{2}\pi\:{such}\:{that}:{f}\left({x}\right)=\left\{{x},\:\mathrm{0}<{x}<\pi\right. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left\{\pi,\:\pi<{x}<\mathrm{2}\pi\right. \\ $$$${show}\:{that}\:{the}\:{fourier}\:{series}\:{for}\:{f}\left({x}\right)\:{in}\:{the}\:{interval}\:\mathrm{0}<{x}<\mathrm{2}\pi\:{is} \\ $$$$\frac{\mathrm{3}\pi}{\mathrm{4}}−\frac{\mathrm{2}}{\pi}\left[{cosx}+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }{cos}\mathrm{3}{x}+\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{2}} }{cos}\mathrm{5}{x}+…\right]−\left[{sinx}+\frac{\mathrm{1}}{\mathrm{2}}{sin}\mathrm{2}{x}+\frac{\mathrm{1}}{\mathrm{3}}{sin}\mathrm{3}{x}+…\right] \\ $$