for n = 0 the term in your series is 0. so i'll assume that n > 0. let

each is integrable on any interval [a, b], because it's continuous almost everywhere on the

interval. let so is integrable on [a, b] for all n.

since by Weierstrass test, is uniformly convergent. so

is integrable on [a, b] and if a = 0 and b = 1, then this gives us: