Express the Fourier transform in terms of sin and cos instead of using exponentials (it's fairly easy to switch between the two, since e^inx + e^-inx = 2cosx and e^inx - e^-inx = 2isinx). Clearly there will be no sin terms since this is an even function (extending it in the logical way.) Now you should get f(x) = pi/2 + 2/pi Sum[((1-(-1)^n)/n^2) cosnx, n=1 to inf] which is the same as the series given in the question. In fact I'm not sure if your transform is correct, but I'm afraid I don't have time to check. I did this sort of stuff last year, so I'm not too sure about the convergence, but since the Fourier coefficients look like 1/n^2 and |cosx| is bounded by 1 then the series is absolutely convergent by comparison to 1/n^2 (i think.) Hope that helps.