3. For each of the following sequences of real-valued functions on R, use the definition to show that { fn (x)}, converges pointwise to the specified f (x) on the given set I; then determine whether or not the convergence is uniform. Use the definition or its negation to justify your conclusions concerning uni- form convergence. 2x (a) {fn (x)}, ;f (x) = 0; I = [0, 1] 1+ nx (b) {fn (x)} cos nx }; f (x) = 0; I = [0, 1]

3. For each of the following sequences of real-valued functions on R, use the definition to show that { fn (x)}, converges pointwise to the specified f (x) on the given set I; then determine whether or not the convergence is uniform. Use the definition or its negation to justify your conclusions concerning uni- form convergence. 2x (a) {fn (x)}, ;f (x) = 0; I = [0, 1] 1+ nx (b) {fn (x)} cos nx }; f (x) = 0; I = [0, 1]

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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(c) Given two sequences (fn)neN, (9n) nEN C C[0, 1] of continuous functions on the closed unit interval [0, 1] defined by nx nx fn(2) = and g(x) = 1+nx²¹ 1+ n²x² Find the limit f and g, respectively of each sequence, if it exists. Which of these sequences converge uniformly on [0, 1]? That is, do f and g belong to C[0, 1] or not?
HG
I need solution :Q7
Find the positive root of (2) using Bisection method in the interval [2.5, 4]. To what value do the iterations converge? [You can use as many iterations as you need to see the limit] f(x)= x³6x² +11x6 = 0 - Show
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PLease solve both a and b