Examples of Fourier series Example 1 :Solution : Fourier series of \[f\left(t\right)=t+\pi,\ -2\pi<t<2\pi\] is given as \[f\left(t\right)=\frac{a_0}{2}+\sum_{n=1}^{\infty}{a_n\cos(nt)}+b_n\sin(nt)\] Where \[a_0=\frac{1}{2\pi}\int_{-2\pi}^{2\pi}f\left(t\right)dt=\frac{1}{2\pi}\int_{-2\pi}^{2\pi}{(t+\pi)dt}=\frac{1}{2\pi}4\pi^2=2\pi\] \[a_n=\frac{1}{2\pi}\int_{-2\pi}^{2\pi}f\left(t\right)dt=\frac{1}{2\pi}\int_{-2\pi}^{2\pi}{\left(t+\pi\right)\cos{\left(nt\right)}dt}=\frac{1}{2\pi}2πsin2nπ]n=sin2nπ]n\] Since \[\sin{\left(2n\pi\right)}=0\ for all n = 1,2, 3,4,5,…\] \[a_n=0, n = 1,2, 3,4,5,…\] Thus And \[b_n=\frac{1}{2\pi}\int_{-2\pi}^{2\pi}f\left(t\right)dt=\frac{1}{2\pi}\int_{-2\pi}^{2\pi}{\left(t+\pi\right)\sin{\left(nt\right)}dt}\] \[b_n=\frac{1}{2\pi}\frac{-4n\pi\cos{\left(2n\pi\right)}+2Sin\left(2n\pi\right)}{n^2},\ n=1,2,3,4,,\ldots\] Since \[\cos{\left(2n\pi\right)}=\left(-1\right)^{2n}=1,\…
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