Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 30A / c / Solution

Insert the uniformly convergent Fourier series of $g$ into the interval formula and integrate termwise. The constant mode contributes $t{g}_0$, while for $k\ne 0$,The sum is exactly the expression from part (b), proving agreement of the transform and Fourier-series formulas.