Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 39C / b / Solution

Insert the Fourier seriesinto the rectangle rule. The finite geometric sum obeysConsequently the quadrature value iswhereasThus the exact Periodic N-point rectangle-rule error is

By the Fourier coefficient decay of an analytic periodic function, there are constants $C>0$ and $0<\rho <1$ such thatIt follows thatThereforewhich is exponential, and hence spectral, convergence.