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Sequences and Series of Functions Chapter 5 Sequences and Series of Functions In this chapter, we define and study the convergence of sequences and series of functions. There are many different ways to define the convergence of a sequence of functions, and different definitions lead to inequivalent types of convergence. We consider here two basic types: pointwise and uniform convergence. 5.1. Pointwise convergence Pointwise convergence defines the convergence of functions in terms of the conver- gence of their values at each point of their domain. Definition 5.1. Suppose that (fn) is a sequence of functions fn : A ! R and f : A ! R. Then fn ! f pointwise on A if fn(x) ! f(x) as n ! 1 for every x 2 A. We say that the sequence (fn) converges pointwise if it converges pointwise to some function f, in which case f(x) = lim fn(x): n!1 Pointwise convergence is, perhaps, the most natural way to define the convergence of functions, and it is one of the most important. Nevertheless, as the following examples illustrate, it is not as well-behaved as one might initially expect. Example 5.2. Suppose that fn : (0; 1) ! R is defined by n f (x) = : n nx + 1 Then, since x =6 0, 1 1 lim fn(x) = lim = ; n!1 n!1 x + 1=n x 57 58 5. Sequences and Series of Functions so fn ! f pointwise where f : (0; 1) ! R is given by 1 f(x) = : x We have jfn(x)j < n for all x 2 (0; 1), so each fn is bounded on (0; 1), but their pointwise limit f is not. Thus, pointwise convergence does not, in general, preserve boundedness. n Example 5.3. Suppose that fn : [0; 1] ! R is defined by fn(x) = x . If 0 ≤ x < 1, n n then x ! 0 as n ! 1, while if x = 1, then x ! 1 as n ! 1. So fn ! f pointwise where ( 0 if 0 ≤ x < 1, f(x) = 1 if x = 1. Although each fn is continuous on [0; 1], their pointwise limit f is not (it is discon- tinuous at 1). Thus, pointwise convergence does not, in general, preserve continuity. Example 5.4. Define fn : [0; 1] ! R by 8 <>2n2x if 0 ≤ x ≤ 1=(2n) 2 fn(x) = 2n (1=n − x) if 1=(2n) < x < 1=n, :> 0 1=n ≤ x ≤ 1. If 0 < x ≤ 1, then fn(x) = 0 for all n ≥ 1=x, so fn(x) ! 0 as n ! 1; and if x = 0, then fn(x) = 0 for all n, so fn(x) ! 0 also. It follows that fn ! 0 pointwise on [0; 1]. This is the case even though max fn = n ! 1 as n ! 1. Thus, a pointwise convergent sequence of functions need not be bounded, even if it converges to zero. Example 5.5. Define fn : R ! R by sin nx f (x) = : n n ! R 0 0 Then fn 0 pointwise on . The sequence (fn) of derivatives fn(x) = cos nx does not converge pointwise on R; for example, 0 − n fn(π) = ( 1) does not converge as n ! 1. Thus, in general, one cannot differentiate a pointwise convergent sequence. This is because the derivative of a small, rapidly oscillating function may be large. Example 5.6. Define fn : R ! R by x2 fn(x) = p : x2 + 1=n If x =6 0, then x2 x2 lim p = = jxj n!1 x2 + 1=n jxj while fn(0) = 0 for all n 2 N, so fn ! jxj pointwise on R. The limit jxj not differentiable at 0 even though all of the fn are differentiable on R. (The fn \round off" the corner in the absolute value function.) 5.2. Uniform convergence 59 Example 5.7. Define f : R ! R by n ( ) x n f (x) = 1 + : n n x Then by the limit formula for the exponential, which we do not prove here, fn ! e pointwise on R. 5.2. Uniform convergence In this section, we introduce a stronger notion of convergence of functions than pointwise convergence, called uniform convergence. The difference between point- wise convergence and uniform convergence is analogous to the difference between continuity and uniform continuity. Definition 5.8. Suppose that (fn) is a sequence of functions fn : A ! R and f : A ! R. Then fn ! f uniformly on A if, for every ϵ > 0, there exists N 2 N such that n > N implies that jfn(x) − f(x)j < ϵ for all x 2 A: When the domain A of the functions is understood, we will often say fn ! f uniformly instead of uniformly on A. The crucial point in this definition is that N depends only on ϵ and not on x 2 A, whereas for a pointwise convergent sequence N may depend on both ϵ and x. A uniformly convergent sequence is always pointwise convergent (to the same limit), but the converse is not true. If for some ϵ > 0 one needs to choose arbitrarily large N for different x 2 A, meaning that there are sequences of values which converge arbitrarily slowly on A, then a pointwise convergent sequence of functions is not uniformly convergent. n Example 5.9. The sequence fn(x) = x in Example 5.3 converges pointwise on [0; 1] but not uniformly on [0; 1]. For 0 ≤ x < 1 and 0 < ϵ < 1, we have n jfn(x) − f(x)j = jx j < ϵ if and only if 0 ≤ x < ϵ1=n. Since ϵ1=n < 1 for all n 2 N, no N works for all x sufficiently close to 1 (although there is no difficulty at x = 1). The sequence does, however, converge uniformly on [0; b] for every 0 ≤ b < 1; for 0 < ϵ < 1, we can take N = log ϵ/log b. Example 5.10. The pointwise convergent sequence in Example 5.4 does not con- verge uniformly. If it did, it would have to converge to the pointwise limit 0, but ( ) 1 fn = n; 2n so for no ϵ > 0 does there exist an N 2 N such that jfn(x) − 0j < ϵ for all x 2 A and n > N, since this inequality fails for n ≥ ϵ if x = 1=(2n). Example 5.11. The functions in Example 5.5 converge uniformly to 0 on R, since j sin nxj 1 jf (x)j = ≤ ; n n n so jfn(x) − 0j < ϵ for all x 2 R if n > 1/ϵ. 60 5. Sequences and Series of Functions 5.3. Cauchy condition for uniform convergence The Cauchy condition in Definition 1.9 provides a necessary and sufficient condi- tion for a sequence of real numbers to converge. There is an analogous uniform Cauchy condition that provides a necessary and sufficient condition for a sequence of functions to converge uniformly. Definition 5.12. A sequence (fn) of functions fn : A ! R is uniformly Cauchy on A if for every ϵ > 0 there exists N 2 N such that m; n > N implies that jfm(x) − fn(x)j < ϵ for all x 2 A: The key part of the following proof is the argument to show that a pointwise convergent, uniformly Cauchy sequence converges uniformly. Theorem 5.13. A sequence (fn) of functions fn : A ! R converges uniformly on A if and only if it is uniformly Cauchy on A. Proof. Suppose that (fn) converges uniformly to f on A. Then, given ϵ > 0, there exists N 2 N such that ϵ jf (x) − f(x)j < for all x 2 A if n > N: n 2 It follows that if m; n > N then jfm(x) − fn(x)j ≤ jfm(x) − f(x)j + jf(x) − fn(x)j < ϵ for all x 2 A; which shows that (fn) is uniformly Cauchy. Conversely, suppose that (fn) is uniformly Cauchy. Then for each x 2 A, the real sequence (fn(x)) is Cauchy, so it converges by the completeness of R. We define f : A ! R by f(x) = lim fn(x); n!1 and then fn ! f pointwise. To prove that fn ! f uniformly, let ϵ > 0. Since (fn) is uniformly Cauchy, we can choose N 2 N (depending only on ϵ) such that ϵ jf (x) − f (x)j < for all x 2 A if m; n > N: m n 2 Let n > N and x 2 A. Then for every m > N we have ϵ jf (x) − f(x)j ≤ jf (x) − f (x)j + jf (x) − f(x)j < + jf (x) − f(x)j: n n m m 2 m Since fm(x) ! f(x) as m ! 1, we can choose m > N (depending on x, but it doesn't matter since m doesn't appear in the final result) such that ϵ jf (x) − f(x)j < : m 2 It follows that if n > N, then jfn(x) − f(x)j < ϵ for all x 2 A; which proves that fn ! f uniformly. 5.4. Properties of uniform convergence 61 Alternatively, we can take the limit as m ! 1 in the Cauchy condition to get for all x 2 A and n > N that ϵ jf(x) − fn(x)j = lim jfm(x) − fn(x)j ≤ < ϵ. m!1 2 5.4. Properties of uniform convergence In this section we prove that, unlike pointwise convergence, uniform convergence preserves boundedness and continuity. Uniform convergence does not preserve dif- ferentiability any better than pointwise convergence. Nevertheless, we give a result that allows us to differentiate a convergent sequence; the key assumption is that the derivatives converge uniformly. 5.4.1.
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