1. Determine whether the following sequences converge pointwise or uniformly on R wherek E N.(a)(b)(c)(d)fk (x)fk (x)=9k (x) ===9k(X) = {{k²,1x²,sin xkX1+kx²if |x| ≤ kif |x|> k.((-1)²) o-) cos(kx)

Answered: Image /qna-images/answer/0077ccfe-c7aa-4f3a-9bed-54b3992384a4.jpg

Answered: fk (x) = fk (x) 9k (x) = { 9k (x) = = =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Consider the sequence of functions {fn(x)} = {n²x(1 – x)"} for a E R and n = = 1,2, 3.... (a) Sketch on the same axis the graphs of f1(x) to f5(x).
6. Consider the sequence of functions n²a4 – sin(nx) fn(x) x € (0, 00) n2x (i) Show that fn converges pointwise and determine the limit. (ii) Use the inequality | sin x| < |x], x €R to show that fn converges uniformly to the above function in (0, 00). (iii) Calculate the limit p² nx4 – sin(nx) lim dx. n2x Justify any possible exchange of the integral with the limit.
2. Determine if the given sequence converges or diverges. If it converges, what is its limit? (e) a, = In(6n³ – 4) – In(3n³ + 5n² – 7n + 8) %3D (f) {cos(nx)}o
Find the set A on which the sequence {fn} converges pointwise. Find the limit function. Choose two (2) items only. 1 (a) fn(r)=r"(1−g") (c) fn(2)= (x+n)² x² + n² (b) fn(x) = 1+x" 1+x²n
(c) Let ƒ : [−1,1] → R be a continuous function. Prove that the sequence (™) defined inductively by 1 Xn+1 = Xn + 2nf ( Xn 1+|xn| x1 = 0, converges. You may use theorems from the lecture as long as you state them explicitly.
Q(A). Let {fn(x)}-1=3: be a sequence of functions + (x – 2)* S. n=1 defined over [2,3]. Show that: (a) fn(x) is meaurable and monotonic increasing for all n.
Short.
(c) A sequence (fn) of differentiable functions such that both (fn) and (fn )converge uniformly but f = lim fn is not differentiable at some point
Q9
nx +n² +x 1. Explore the uniform convergence of the functional sequence f,(x) = n² +x? on the set E = [0,1].
(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?
1. Let (xn) be an increasing sequence of positive real numbers and define the sequence (yn) as follows: For each natural number n, Xn Yn 1+ xn Prove that (yn) is convergent. What can you say about the limit of the sequence (yn)?

SEE MORE QUESTIONS

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

fk (x) = fk (x) 9k (x) = { 9k (x) = = = sin x k X 1+kx² if |x ≤ k if |x|> k. ((-1²) c -) cos(kx)