First few truncated Fourier series for the 2-periodic odd extension of the function f(x)=x initially defined on 0<x<1.

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2*sin(Pi*x)/Pi

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2*sin(Pi*x)/Pi-sin(2*Pi*x)/Pi+2/3*sin(3*Pi*x)/Pi

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2*sin(Pi*x)/Pi-sin(2*Pi*x)/Pi+2/3*sin(3*Pi*x)/Pi-1/2*sin(4*Pi*x)/Pi+2/5*sin(5*Pi*x)/Pi

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2*sin(Pi*x)/Pi-sin(2*Pi*x)/Pi+2/3*sin(3*Pi*x)/Pi-1/2*sin(4*Pi*x)/Pi+2/5*sin(5*Pi*x)/Pi-1/3*sin(6*Pi*x)/Pi+2/7*sin(7*Pi*x)/Pi