Fourier series and convergence
General · Mathematics
Study notes
Q: Find the Fourier series of f(x) = x on (-π,π). Odd: aₙ = 0. bₙ = (2/π)∫₀^πx sin(nx)dx. Integrate: bₙ = (2/π)[-x cos(nx)/n|₀^π+∫cos(nx)/n] = 2(-1)^(n+1)/n. f(x) = Σ2(-1)^(n+1)sin(nx)/n. At x = π/2: π/2 = 2(1-1/3+1/5-...): Leibniz π/4! ✓! (Gibbs: overshoot at jumps!)