The following function is periodic between -π and π:
f(x) = |x|
Find the Coefficients of the Fourier series and, by examining the Fourier series at x=π or otherwise, determine:
1 + 1/32 + 1/52 + 1/72 ... = Σ∞j=1 1/(2j - 1)2
f(x) = a0/2 + ∑∞n=1 ancos(nx) + bn sin(nx)
a0 = 1/π ∫π-π f(x) dx
an = 1/π ∫π-π f(x) cos(nx) dx
bn = 1/π ∫π-π f(x) sin(nx) dx
The Attempt at a Solution
So I've found the coefficients:
a0 = π
an = -2(1 - (-1)n)/πn2
bn = 0 (as even function)
and so f(x) = π/2 + ∑-2(1 - (-1)n)/πn2 cos(nx)
I don't know how to do the last bit however, I don't really understand if I'm meant to come out with a number or something...
Any help would be greatly appreciated thank you.