Show that the characteristic function of snnbspcan be

Question

Let  be a sequence of IID random variables with finite mean, µ, and let S_n be the sequence of sample means,

(a) Show that the characteristic function of S_n can be written as

 (b) Use Taylor's theorem to write the characteristic function of the X_K

In so doing, you have proved that the distribution of the sample mean is that of a constant in the limit as  Thus, the sample mean converges in distribution.