Question 9.6: Determine if the signal x[n] = ∑m=0 ^∞ Xm cos(mω0n), ω0 = 2π......

The signal x[n] consists of the sum of a constant X_0 and cosines of frequency

m\omega_{0}={\frac{2\pi m}{N_{0}}}\qquad m=1,2,\cdots.

The periodicity of x[n] depends on the periodicity of the cosines. According to the frequency of the cosines, they are periodic of fundamental period N_0. Thus x[n] is periodic of fundamental period N_0, indeed

x[n+N_{0}]=\sum_{m=0}^{\infty}X_{m}\cos(m\omega_{0}(n+N_{0}))=\sum_{m=0}^{\infty}X_{m}\cos(m\omega_{0}n+2\pi m)=x[n].