Weierstrass M-Test

Real Analysis · Axiom Academy

A single height cap on every term — when the caps add up to something finite, the whole series of functions converges uniformly. Give each function f_n a single number M_n that caps its height everywhere on S : for all x . The cap can't depend on x — it must hold for the whole set at once. If the caps themselves add up to a finite total, that total controls the entire series. A height cap that works for every x The caps add to a finite total uniform convergence Take on the whole real line. Since for every x , each term is trapped: . So the cap is — and the watch below shows the wiggling term staying inside that flat band no matter how far out in x you look. oscillates forever as x runs over . The bound uses no special x — it is uniform on . Caps that add to a finite total, so the M-Test fires: converges uniformly on . 3. Why It Works: the Tail Goes to Zero Uniform convergence means the partial sums are uniformly Cauchy : a tail block is small for every x at once. The caps deliver exactly that — and the watch below shows the cap-tail bar collapsing to 0 as n grows, dragging the function tail down with it. Conclusion. The bound does not mention x , so one N works across all of S . By the Cauchy criterion, converges uniformly. 4. Why Uniform Convergence Is Worth It

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