Fourier Series Visualization

Square Wave: Exhibits odd symmetry ($a_k = 0$) and half-wave symmetry, meaning only odd harmonics ($k=1, 3, 5...$) exist. Notice the Gibbs phenomenon: the ~9% overshoot at the discontinuities never disappears, even as $N \to \infty$. The coefficients decay as $1/k$.

$$x(t)\approx \frac{4}{\pi }\sum _{k=1,3,5...}^N\frac{1}{k}\sin (kt)$$