Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 2 / 17 / a

Every harmonic in this Fourier series has period $2L$. Since the first-harmonic coefficient is nonzero, the fundamental period cannot be smaller: translating by a period $T$ must multiply that nonzero complex coefficient by $e^{i\pi T/L}=1$. Thus, taking $L>0$,