# Counterexamples on Uniform Convergence Sequences, Series, Functions, and Integrals von Bourchtein, Andrei (eBook)

• Erscheinungsdatum: 23.01.2017
• Verlag: Wiley
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## Counterexamples on Uniform Convergence

A comprehensive and thorough analysis of concepts and results on uniform convergence

Counterexamples on Uniform Convergence: Sequences, Series, Functions, and Integrals presents counterexamples to false statements typically found within the study of mathematical analysis and calculus, all of which are related to uniform convergence. The book includes the convergence of sequences, series and families of functions, and proper and improper integrals depending on a parameter. The exposition is restricted to the main definitions and theorems in order to explore different versions (wrong and correct) of the fundamental concepts and results.

The goal of the book is threefold. First, the authors provide a brief survey and discussion of principal results of the theory of uniform convergence in real analysis. Second, the book aims to help readers master the presented concepts and theorems, which are traditionally challenging and are sources of misunderstanding and confusion. Finally, this book illustrates how important mathematical tools such as counterexamples can be used in different situations.

The features of the book include:

- An overview of important concepts and theorems on uniform convergence

- Well-organized coverage of the majority of the topics on uniform convergence studied in analysis courses

- An original approach to the analysis of important results on uniform convergence based\ on counterexamples

- Additional exercises at varying levels of complexity for each topic covered in the book

- A supplementary Instructor's Solutions Manual containing complete solutions to all exercises, which is available via a companion website

Counterexamples on Uniform Convergence: Sequences, Series, Functions, and Integrals is an appropriate reference and/or supplementary reading for upper-undergraduate and graduate-level courses in mathematical analysis and advanced calculus for students majoring in mathematics, engineering, and other sciences. The book is also a valuable resource for instructors teaching mathematical analysis and calculus.

ANDREI BOURCHTEIN, PhD, is Professor in the Department of Mathematics at Pelotas State University in Brazil. The author of more than 100 referred articles and five books, his research interests include numerical analysis, computational fluid dynamics, numerical weather prediction, and real analysis. Dr. Andrei Bourchtein received his PhD in Mathematics and Physics from the Hydrometeorological Center of Russia.

LUDMILA BOURCHTEIN, PhD, is Senior Research Scientist at the Institute of Physics and Mathematics at Pelotas State University in Brazil. The author of more than 80 referred articles and three books, her research interests include real and complex analysis, conformal mappings, and numerical analysis. Dr. Ludmila Bourchtein received her PhD in Mathematics from Saint Petersburg State University in Russia.

ANDREI BOURCHTEIN, PhD, is Professor in the Department of Mathematics at Pelotas State University in Brazil. The author of more than 100 referred articles and five books, his research interests include numerical analysis, computational fluid dynamics, numerical weather prediction, and real analysis. Dr. Andrei Bourchtein received his PhD in Mathematics and Physics from the Hydrometeorological Center of Russia.

LUDMILA BOURCHTEIN, PhD, is Senior Research Scientist at the Institute of Physics and Mathematics at Pelotas State University in Brazil. The author of more than 80 referred articles and three books, her research interests include real and complex analysis, conformal mappings, and numerical analysis. Dr. Ludmila Bourchtein received her PhD in Mathematics from Saint Petersburg State University in Russia.

## Produktinformationen

Format: ePUB
Seitenzahl: 272
Erscheinungsdatum: 23.01.2017
Sprache: Englisch
ISBN: 9781119303428
Verlag: Wiley
Größe: 22414 kBytes
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## Counterexamples on Uniform Convergence

List of Examples

Chapter 1. Conditions of Uniform Convergence

Example 1 . A function f ( x , y ) defined on converges pointwise on X as y approaches y 0, but this convergence is nonuniform on X

A sequence of functions converges (pointwise) on a set, but this convergence is nonuniform.

A series of functions converges (pointwise) on a set, but this convergence is nonuniform.

Example 2 . A series of functions converges on X and a general term of the series converges to zero uniformly on X , but the series converges nonuniformly on X .

Example 3 . A sequence of functions converges on X and there exists its subsequence that converges uniformly on X , but the original sequence does not converge uniformly on X .

Example 4 . A function f ( x , y ) defined on converges on ( a , b ) as y approaches y 0 and this convergence is uniform on any interval , but the convergence is nonuniform on ( a , b ).

A sequence of functions defined on ( a , b ) converges uniformly on any interval , but the convergence is nonuniform on ( a , b )

A series of functions converges uniformly on any interval , but the convergence is nonuniform on ( a , b ).

Example 5 . A sequence converges on X , but this convergence is nonuniform on a closed interval .

A series converges on X , but this series does not converge uniformly on a closed subinterval .

A function f ( x , y ) defined on has a limit for , but f ( x , y ) converges nonuniformly on a subinterval .

Example 6 . A sequence converges on a set X , but it does not converge uniformly on any subinterval of X .

A series converges on X , but it does not converge uniformly on any subinterval of X .

Example 7 . A series converges uniformly on an interval, but it does not converge absolutely on the same interval.

Example 8 . A series converges absolutely on an interval, but it does not converge uniformly on the same interval.

Example 9 . A series converges absolutely and uniformly on [ a , b ], but the series does not converge uniformly on [ a , b ].

Example 10 . A series converges absolutely and uniformly on X , but there is no bound of the general term u n ( x ) on X in the form , such that the series converges.

Example 11 . A sequence f n ( x ) converges uniformly on X to a function f ( x ), but does not converge uniformly on X to f 2( x ).

Sequences f n ( x ) and g n ( x ) converge uniformly on X , but f n ( x ) g n ( x ) does not converge uniformly on X .

Example 12 . Sequences f n ( x ) and g n ( x ) converge nonuniformly on X to f ( x ) and g ( x ), respectively, but converges to uniformly on X .

Example 13 . A sequence converges uniformly on X , but f n ( x ) diverges on X .

A sequence converges uniformly on X , but f n ( x ) diverges on X .

A sequence converges uniformly on X and f n ( x ) converges on X , but the convergence of f n ( x ) is nonuniform.

Example 14 . A sequence converges uniformly on X to 0, but neither f n ( x ) nor g n ( x ) converges to 0 on X .

Example 15 . A sequence f n ( x ) converges uniformly on X to a function f ( x ), , , , but do

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