18.100A: Complete Lecture Notes Lecture 23: Pointwise and Uniform Convergence of Sequences of Functions Sequences of Function Power Series Remark 1. Power series motivate the general discussion of sequences of functions. Definition 2 (Power series) A power series about x0 is a series of the form ∞ X am (x − x0 )m . m=0 Theorem 3 Suppose R = lim |am |1/n m→∞ exists, and let p= Then, P  1 R>0 R ∞ R = 0. am (x − x0 )m converges absolutely if |x − x0 | < p and diverges if |x − x0 | > p. Definition 4 (Radius of Convergence) In the above theorem, we define p to be the radius of convergence. Proof : We have lim |am (x − x0 )m |1/m = R|x − x0 |, n→∞ and the theorem follows by the Root test. P Suppose am (x − x0 )m is a power series with radius of convergence p. Furthermore, define f : (x0 − p, x0 + p) → R such that ∞ X f (x) := am (x − x0 )m . m=0 Then, f is a limit of a sequence of functions f (x) = lim fn (x), n→∞ 1 for x ∈ (x0 − p, x0 + p) and where n X fn (x) = am (x − x0 )m . m=0 Example 5 For example, we have ∞ X 1 = xm . 1 − x m=0 f (x) = Question 6. This concept begs a number of questions: 1. Is f continuous? 2. Is f differentiable, and does f 0 = limn→∞ fn0 ? 3. If 1. is true, does Z b b Z f = lim n→∞ a fn ? a These questions will be the key motivator for the last section of this course. Pointwise and Uniform Convergence We now consider a setting far more general than power series. Definition 7 (Pointwise Convergence) For n ∈ N, let fn : S → R. Let f : S → R. We say that {fn } converges pointwise to f if for all x ∈ S, lim fn (x) = f (x). n→∞ Let’s consider some examples. 1. Let fn (x) = xn on [0,1]. Then,  0 x ∈ [0, 1) lim fn (x) = . n→∞ 1 x = 1 Thus, {fn } converges to the above pointwise function. Hence, notice that a sequence of continuous functions may not converge pointwise to a continuous function! Pn 2. Let fn (x) = m=0 xm for x ∈ (−1, 1). Then, lim fn (x) = lim n→∞ n→∞ n X m=0 xm = 1 . 1−x Hence, pointwise, this sequence converges to its power series (see the above example). 3. Let fn (x) : [0, 1] → R be defined by  1   4n2 x x ∈ 0, 2n   1 1 fn (x) = 4n − 4n2 x x ∈ 2n ,n .    0 x ∈ n1 , 1 We can picture this sequence (on the next page) 2 Then, limn→∞ fn (0) = limn→∞ 0 = 0. Let x ∈ (0, 1]. Let N ∈ N such that 1 N < x. Then, for all n ≥ N , fn (x) = 0. Therefore, {fn (x)} = f1 (x), . . . , fN −1 (x), 0, 0, 0, . . . . Hence, limn→∞ fn (x) = 0 for all x ∈ [0, 1]. Thus, {fn } converges pointwise to f (x) = 0 on [0, 1]. Definition 8 (Uniform Convergence) For n ∈ N, let fn : S → R, and let f : S → R. Then, we say fn converges to f uniformly or converges uniformly to f if ∀ > 0 ∃M ∈ N such that for all n ≥ M ∀x ∈ S, |fn (x) − f (x)| < Theorem 9 If fn : S → R, f : S → R, and fn → f uniformly, then fn → f pointwise. Proof : Let c ∈ S and let > 0. Then, fn → f uniformly implies that there exists M0 ∈ N such that for all n ≥ M, ∀x ∈ S, |fn (x) − f (x)| < . Choose M = M0 . Then, ∀n ≥ M , |fn (c) − f (c)| < . Thus, limn→∞ fn (c) = f (c) for all c ∈ S, and therefore fn → f pointwise. 3