Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 18 / b / ii / Solution

If $\ell =mL$ with integer $m>0$, then $k_n\ell =2\pi nm$. All sine phase factors vanish and all cosine phase factors equal one. Thereforefor the periodic extension, and every Fourier coefficient is unchanged.