MAT242 Summary of Chapter

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MAT242 2011-2012 Summary of Chapter 1 I. Definitions 1. We say that a sequence of functions (fn)n converges pointwise on X to a function f if for every x ? X, for every > 0 there exists N ? N such that if n ≥ N then |fn(x)? f(x)| ≤ . We say that a series of functions ∑∞ n=1 fn converges pointwise on X to a function f if for all x ? X the partial sums ∑n k=1 fk converge to f(x). 2. We say that a sequence of functions (fn)n converges uniformly on X to a function f if for every > 0 there exists N ? N such that if n ≥ N then |fn(x) ? f(x)| ≤ for all x ? X, or in other words supx?X |fn(x)? f(x)| ? 0 as n?∞. We say that a series of functions ∑∞ n=1 fn converges uniformly on X to a function f if the partial sums ∑n k=1 fk converge to f(x) uniformly on x. 3. We say that a series ∑ un converges normally on X if the series ∑ supx?X |un(x)| converges.

• ∑n


• then


• partial sums


• xn does


• x0 ?


• fn


• ∑∞


• sums


• k2 ≤


• normal convergence

Sujets

• Series (mathematics)
• Normal convergence

Informations

• Publié par: pefav
• Nombre de lectures: 9
• Langue: English

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MAT242 2011-2012

Summary of Chapter 1

I. Definitions 1. Wesay that a sequence of functions(f n ) n convergespointwiseonXto a functionfif for every x∈X, for every >0there existsN∈Nsuch that ifn≥Nthen|f n (x)−f(x)| ≤. P ∞ We say that a series of functionsf n convergespointwiseonXto a functionfif for all n=1 P n x∈Xconverge tothe partiaf(x). l sums =1 f k k 2. Wesay that a sequence of functions(f n ) n convergesuniformlyonXto a functionfif for every  >0there existsN∈Nsuch that ifn≥Nthen|f n (x)−f(x)| ≤for allx∈X, or in other dssup wor x∈X |f n (x)−f(x)| →0asn→ ∞. P ∞ We say that a series of functionsf n convergesuniformlyonXto a functionfif the n=1 P n nverge tof(x)uniformly onx. partial sums k=1 f k co P P eriessupx)|converges. 3. Wesay that a seriesu n convergesnormallyonXif the s x∈X |u n ( 4. For allnletf n :A→R. ThesetD={x∈A|(f n (x)) n converges}is calleddomain of convergence. II. Theorems 1. Let(f n ) n be a sequence of continuous functions onXand letf n converge tofuniformly, then fcontinuous. 2. Let(f n ) n be a sequence of continuous functions on the interval[a, b]that converges uniformly R RR b bb tof. Thenlim n→∞ f n (x)dx= lim n→∞ f n (x)dx=f(x)dx. a aa P 3. Let(f n ) n be a sequence of continuous functions on the interval[a, b]such thatf n converges R PP R b ∞ ∞ b on[a, b]. Thenf n (x)dx=f n (x)dx. a n=1n=1 a 4. Let(f n ) n be a sequence of continuous, differentiable functions with a continuous derivative (i.e. 10 f∈ C(I)) on an intervalI= [f n a, b]such that (1)f n (x 0 )→y 0 , for somex 0 ∈I; (2) n →g uniformly; R x then if we letf(x) =y 0 +g(t)dt, thenfis differentiable,(f n ) n converges tofuniformly and x 0 df n d 0 g(x) = lim n→∞ = (lim n→∞ f n ) =f(x). dx dx 0 0 df→guniformly ofis differentiable andf=g 5. Iff n →funiformly, an n n an intervalIthen onI. 1 6. Foralln∈Nletu n :I→RSuppose that (1) for allbe a function.n∈Nu n ∈ C(I); (2) P P 0 e seriesc rgesuniformly onI thu n onve ;(3) there existsx 0 ∈Isuch that the seriesu n (x 0 ) P P ∞ ∞ 0 0 converges; then(u n ) =u. n=0n=0 n III. PointwiseConvergence 1. Howdo we show that a sequence of functionsf n converges pointwise onX? STEP 1:We find a limit functionf. Then (i) Bydefinition. We are given anx∈X, we are given >0, then wefindN∈N(that can depend on andx) and wecheckthat ifn≥Nthen|f n (x)−f(x)| ≤. sinnx √ EX.The sequence of functionsf n (x) =converges pointwise tof(x) = 0onR.Let n |sinnx| 1 sinnx1 x∈R.Let >0.ChooseN > 2 . Forn > Nwe have| −0| ≤≤ ≤ √ √√  n n n 1 √ ≤, and hencef n →fpointwise. N (ii) Bydefinition version 2. Weshowthat for allx∈Xthe number sequence(f n (x)) n converges tof(x)is that. That |f n (x)−f(x)| →0. sinnx √ EX.The sequence of functionsf n (x) =converges pointwise tof(x) = 0onR.Let n |sinnx| sinnx1 √ √√ x∈Rhave. We| −0| ≤≤ →0, asn→ ∞and hencef n →fpointwise. n nn