Question

Find the closed form of the Fourier series of f(x) with integrals simplified and evaluated. Use Octave or Matlab to plot a graph of the first 3 non-zero partial sums. Period is 2pi

f(x)=x² (-pi<x<pi)

Answer 1

The function f(x) = x² is even. So it will have fourier cosine series

where

where I have used the integral property of even fucntion

and

Integrate by parts to obtain

Integrate by parts one more time

But sin(n Pi) = 0

Therefore the Fourier series becomes

The first 3 non zero terms are

Plotting this is mathematica gives