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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.

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Series (mathematics) Normal convergence

MAT242 2011-2012

Summary of Chapter 1

I. Deﬁnitions 1. Wesay that a sequence of functions(fn)nconvergespointwiseonXto a functionfif for every x∈X, for every >0there existsN∈Nsuch that ifn≥Nthen|fn(x)−f(x)| ≤. P ∞ We say that a series of functionsfnconvergespointwiseonXto a functionfif for all n=1 P n x∈Xconverge tothe partiaf(x). l sums=1fk k 2. Wesay that a sequence of functions(fn)nconvergesuniformlyonXto a functionfif for every  >0there existsN∈Nsuch that ifn≥Nthen|fn(x)−f(x)| ≤for allx∈X, or in other dssup worx∈X|fn(x)−f(x)| →0asn→ ∞. P ∞ We say that a series of functionsfnconvergesuniformlyonXto a functionfif the n=1 P n nverge tof(x)uniformly onx. partial sumsk=1fkco P P eriessupx)|converges. 3. Wesay that a seriesunconvergesnormallyonXif the sx∈X|un( 4. For allnletfn:A→R. ThesetD={x∈A|(fn(x))nconverges}is calleddomain of convergence. II. Theorems 1. Let(fn)nbe a sequence of continuous functions onXand letfnconverge tofuniformly, then fcontinuous. 2. Let(fn)nbe a sequence of continuous functions on the interval[a, b]that converges uniformly R RR b bb tof. Thenlimn→∞fn(x)dx= limn→∞fn(x)dx=f(x)dx. a aa P 3. Let(fn)nbe a sequence of continuous functions on the interval[a, b]such thatfnconverges R PP R b∞ ∞b on[a, b]. Thenfn(x)dx=fn(x)dx. a n=1n=1a 4. Let(fn)nbe a sequence of continuous, diﬀerentiable functions with a continuous derivative (i.e. 10 f∈ C(I)) on an intervalI= [f na, b]such that (1)fn(x0)→y0, for somex0∈I; (2)n→g uniformly; R x then if we letf(x) =y0+g(t)dt, thenfis diﬀerentiable,(fn)nconverges tofuniformly and x0 dfnd0 g(x) = limn→∞= (limn→∞fn) =f(x). dx dx 0 0 df→guniformly ofis diﬀerentiable andf=g 5. Iffn→funiformly, annn an intervalIthen onI. 1 6. Foralln∈Nletun:I→RSuppose that (1) for allbe a function.n∈Nun∈ C(I); (2) P P 0 e seriesc rgesuniformly onI thunonve ;(3) there existsx0∈Isuch that the seriesun(x0) P P ∞ ∞ 0 0 converges; then(un) =u. n=0n=0n III. PointwiseConvergence 1. Howdo we show that a sequence of functionsfnconverges 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|fn(x)−f(x)| ≤. sinnx √ EX.The sequence of functionsfn(x) =converges pointwise tof(x) = 0onR.Let n |sinnx| 1 sinnx1 x∈R.Let >0.ChooseN >2. Forn > Nwe have| −0| ≤≤ ≤ √ √√  nn n 1 √ ≤, and hencefn→fpointwise. N (ii) Bydeﬁnition version 2. Weshowthat for allx∈Xthe number sequence(fn(x))nconverges tof(x)is that. That |fn(x)−f(x)| →0. sinnx √ EX.The sequence of functionsfn(x) =converges pointwise tof(x) = 0onR.Let n |sinnx| sinnx1 √ √√ x∈Rhave. We| −0| ≤≤ →0, asn→ ∞and hencefn→fpointwise. n nn

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