Fourier series: x + x²

f(x) = x + x² on (−π, π). Split into the odd part x and the even part x².

Coefficients

a₀ = (1/π) ∫_{−π}^{π} (x + x²) dx = (1/π)(2π³/3) = 2π²/3
aₙ = (1/π) ∫ (x + x²) cos nx dx = (1/π) ∫ x² cos nx dx = 4(−1)ⁿ/n²
bₙ = (1/π) ∫ (x + x²) sin nx dx = (1/π) ∫ x sin nx dx = 2(−1)ⁿ⁺¹/n

(x cos nx and x² sin nx are odd, so they integrate to zero.)

Series

x + x² = π²/3 + Σ_{n=1}^∞ [ 4(−1)ⁿ cos nx / n² + 2(−1)ⁿ⁺¹ sin nx / n ]
       = π²/3 − 4cos x + cos 2x − (4/9)cos 3x + ...
         + 2 sin x − sin 2x + (2/3) sin 3x − ...