University:
Massachusetts Institute of Technology
Course:
18.100C | Real Analysis
Academic year:

2024
Views:

425

Pages:

1
Author:

Raina Ballard

18.100C Lecture 16 Summary Pointwise convergence. Examples. Uniform convergence. Theorem 16.1 (Cauchy convergence criterion). A sequence of functions fn : X → R is uniformly convergent if and only if the following holds. For every E > 0 there is an N such that if m, n ≥ N then |fn (x) − fm (x)| < E for all x. Uniform convergence of series of functions. ∞ Corollary 16.2. (Weierstrass criterion) Let n=0 fn be a series of functions. Suppose that there are constants Mn such that |fn (x)| ≤ Mn for all n, x, and ∞ ∞ such that n=0 Mn converges. Then n=0 fn converges uniformly. ∞ Corollary 16.3. Let n=0 an xn be a power series with radius of convergence ρ > 0. Then that series converges uniformly on any interval [−r, r] with r < ρ. Theorem 16.4. If (fn ) are continuous functions converging uniformly towards f , then f is again continuous. ∞ Corollary 16.5. Let f (x) = n=0 an xn be a power series with radius of con vergence ρ > 0. Then f is continuous on (−ρ, ρ).