Sum of 1/n^4 (Fourier Series & Parseval's Theorem)

Sum of 1/n^4 by using Fourier Series and Parseval's Theorem, Fourier coefficients from bprp:    • how to get the Fourier series coefficients...   Sum of 1/n^2 from Peyam:    • Sum of 1/n^2   Sum of 1/n^2 from Max:    • Proof by intuition done by Leonhard Euler,...   Sum of 1/n^2 from bprp:    • a spectacular solution to 1+1/2^2+1/3^2+.....   Unfortunately sum of 1/n^3 doesn't have a nice exact answer (although it's called the Apery's constant, which is the zeta(3)):    • Sum of 1/n^3, Believe In Integrals