Fourier cosine series correspondence for f(x)= x, o < x < pie given by x ~ pie / 2 - 4/n, E infinity on top and n=1 on bottom. cos (an-1)/x / (2n-1)squared, (0 < x < Pie).

Question

Explain why this correspondence is actually an equality for 0 is less than or equal to x and x is less than or equal to Pie. Then explain how we can write

{x} = Pie /2 - 4 / n E infinity on top and n=1 on bottom

cos (2n - 1)x/ (2n-1) squared, (-n is less than or equal to x and x is less than or equal to Pie)