Fourier Series?

Question

How can we find the Fourier Series for f(x) = x^3 without intergrating to find the coefficients, if it is known that f(x) = f(x + 2pi) = x^4 , -pi < x < pi, is (pi^4)/5 + 8pi^2(Sum from 1 to Infinity of: [(-1)^n/(n^2)] cos nx) - 48 (Sum from 1 to Infinity of: [(-1)^n/(n^4)] cos nx)?

Answer 1

If you know the series for f(x) = x^4, then differentiate term by term to get the series for f'(x) = 4x^3. then divide by 4.