8. Series of Functions & Fourier

When series of functions converge uniformly — plus Fourier series and the Gibbs phenomenon.

• The Weierstrass M-Test — If a majorant $M_n$ dominates every term at all $x$ and $\sum M_n$ converges, the series converges uniformly — here $\left|\frac{\sin(nx)}{n^2}\right| \le \frac{1}{n^2} =: M_n$ and $\sum M_n = \frac{\pi^2}{6}$, so the partial-sum error band tightens across the whole $x$-interval at once.
• Dini's Theorem — On a compact set, a monotone sequence of continuous functions converging pointwise converges uniformly — a finite subcover of the open sets $E_n$ grabs them all at once. The twin tool of the M-test.
• The Ringing That Never Dies — Layer sine waves onto a square wave. The ringing beside each jump never dies out, no matter how large $N$ gets.
• Fejér's Theorem — The Cesàro means $\sigma_N$ of Fourier partial sums converge uniformly for continuous functions — because the Fejér kernel is an approximate identity. Overlay the ringing $S_N$ with the smooth $\sigma_N$.
• The Gibbs Phenomenon — Why the bump beside the jump never dies — rescaling by $t = 2Nx$ collapses $S_N$ onto $(2/\pi)\mathrm{Si}(t)$, and the first peak tends to $(2/\pi)\mathrm{Si}(\pi) \approx 1.179$. The overshoot is about 8.95% of the jump.