Finn Fourierrekken til f(x)=xf(x)=xf(x)=x på (−π,π)(-\pi,\pi)(−π,π).
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xxx er odde, så an=0a_n=0an=0. bn=2π∫0πxsinnx dx\displaystyle b_n=\frac{2}{\pi}\int_0^\pi x\sin nx\,dxbn=π2∫0πxsinnxdx. Delvis integrasjon: ∫0πxsinnx dx=[−xcosnxn]0π+1n∫0πcosnx dx=−πcosnπn=π(−1)n+1n\displaystyle \int_0^\pi x\sin nx\,dx=\left[-\frac{x\cos nx}{n}\right]_0^\pi+\frac{1}{n}\int_0^\pi\cos nx\,dx=-\frac{\pi\cos n\pi}{n}=\frac{\pi(-1)^{n+1}}{n}∫0πxsinnxdx=[−nxcosnx]0π+n1∫0πcosnxdx=−nπcosnπ=nπ(−1)n+1. Så bn=2(−1)n+1n\displaystyle b_n=\frac{2(-1)^{n+1}}{n}bn=n2(−1)n+1 og f(x)=∑n=1∞2(−1)n+1nsinnx\displaystyle f(x)=\sum_{n=1}^\infty\frac{2(-1)^{n+1}}{n}\sin nxf(x)=n=1∑∞n2(−1)n+1sinnx.
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