- #1

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## Homework Statement

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 + 1/3

^{2}+ 1/5

^{2}+ 1/7

^{2}... = Σ

^{∞}

_{j=1}1/(2j - 1)

^{2}

## Homework Equations

f(x) = a

_{0}/2 + ∑

^{∞}

_{n=1}a

_{n}cos(nx) + b

_{n}sin(nx)

a

_{0}= 1/π ∫

^{π}

_{-π}f(x) dx

a

_{n}= 1/π ∫

^{π}

_{-π}f(x) cos(nx) dx

b

_{n}= 1/π ∫

^{π}

_{-π}f(x) sin(nx) dx

## The Attempt at a Solution

So I've found the coefficients:

a

_{0}= π

a

_{n}= -2(1 - (-1)

^{n})/πn

^{2}

b

_{n}= 0 (as even function)

and so f(x) = π/2 + ∑-2(1 - (-1)

^{n})/πn

^{2}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.