Step 1: Find the Fourier series of the square wave f(x) = +1 for 0 < x < π and −1 for −π < x < 0, period 2π. Step 2: The function is odd, so all cosine coefficients vanish: a_n = 0. Only sine terms survive. Step 3: b_n = (1/π)∫₋π^π f(x)sin(nx)dx = (2/π)∫_0^π sin(nx)dx (using oddness). Step 4: ∫_0^π sin(nx)dx = [−cos(nx)/n]_0^π = (1 − cos nπ)/n = (1 − (−1)^n)/n. Step 5: For even n this is zero; for odd n it is 2/n. So b_n = 4/(nπ) for odd n, 0 for even n. Step 6: f(x) = (4/π)[sin x + (sin 3x)/3 + (sin 5x)/5 + …]. Result: The square wave is an infinite sum of odd harmonics with amplitudes 4/(nπ) — the Gibbs overshoot near the jumps never fully disappears.