Impulse function?

Question

Use the exponential Fourier series to express periodic impulse function δT = Σδ(t-nT) for n = -∞ to ∞ in terms of exponential Fourier Series Cn.

Answer 1

By definition Cn = (1/T)*integral[within T](f(t)*exp(-jnwt)*dt),
and since f(t) = δ(t) it follows Cn = 1/T. Therefore:
δT = Σ Cn*exp(jnwt) = (1/T)*Σ exp(jnwt) = (1/T)*Σ cos(nwt) [for n = -∞ to ∞] = (1/T) + (2/T)*Σ cos(nwt) [for n = 1 to ∞].