Question

Find the Fourier series expansion of the function f (x) = x –x² ​ (-π,π)

Answer 1

I need use Fourier series of f(x)=x2+x ,x∈(−π,π)

to prove that ∑n≥11n2=π26.

I calculated the Fourier series: x2+x=π23+∑n≥14n2(−1)ncosnx−2n(−1)nsinnx.

And could not find any −π<x<π that could solve my problem.

I double-checked my calculations and could not find any problem with the series, could you please give me some hint about my error ?

I know that we can calculate ∑n≥11n2=π26 by using the Fourier expansion of f(x)=x