Real Analysis IIKindly find rough examples as guide in photo 2. 1. Let fn (0, ∞) → R where fn(x)(0, ∞) → R where f(x) = 0.=nxfor all ne N and let f(a) Graph f1, f2, f3, and f on the same axes.(b) Prove that fn converges pointwise to f on (0, ∞).(c) Does fn converge uniformly to f on (0, ∞o)? Prove your answer.(d) Does fn converge uniformly to f on [1, ∞)? Prove your answer. (0,0)Ex.Let S = [0, 1] x fn(x) = x^ *nexo & 0≤x²1or 5 to f(x) = { 1 + x = 1Jaim: {fr} Conv. ptwise(₁,1)f(x)=x² = x to ¼/a ft = (1/2)fa (x) = x²fa= 1/4f3 (x)=x²9 f¶ (x) = .=XlimSo txES, 200limmEanx"limBf Suppose 0≤x²1 - Then apo fn(x) = (im X"=0Cimlim1^ ==Suppose X=1. Then2-01-14sao fn(x) =fn (x) exists,S₁ = 90 x°S₂ = 96x° +9₁x = 90 +9₁ xS3 = ax +9₁ x + 9₂x²= 90+3 = ½Does {fr} Conv. pluise on s? nfShow of = 10Sofn (x) dx = Area of A = 1/2 ² 1/0 +0=1but So f(x) dx=0 limS = [0, 1] for 122, define fn (x) = ['n³x if 0≤x≤/n2-n²(x - ²) 2 % ≤ x ≤-0 i 3/ / LX ≤Ifn15-01-din fLim So fm (x) dx # Sr f(x) dx

Given: A function fn: 0,∞→R where fnx=1nx∀n∈N and f(x)=0. To prove : a) Plot the graph of…

1. Let fn: (0, ∞) → R where fn(x) (0, ∞)→ R where f(x) = 0. = nx for all ne N and let ƒ : (a) Graph f1, f2, f3, and f on the same axes. (b) Prove that fn converges pointwise to f on (0, ∞). (c) Does fn converge uniformly to f on (0, ∞o)? Prove your answer.

1. Let fn: (0, ∞) → R where fn(x) (0, ∞)→ R where f(x) = 0. = nx for all ne N and let ƒ : (a) Graph f1, f2, f3, and f on the same axes. (b) Prove that fn converges pointwise to f on (0, ∞). (c) Does fn converge uniformly to f on (0, ∞o)? Prove your answer.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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