Exercises of Functions Series E-Book

Simone Malacrida

• Herausgeber: Simone Malacrida
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2022

Beschreibung

In this book exercises are carried out regarding the following mathematical topics:
successions of functions of real variable
series of functions of real variable
diversity of the concepts of convergence of successions and series
Initial theoretical hints are also presented to make the performance of the exercises understood.

Leseprobe

Table of Contents

"Exercises of Functions Series"

INTRODUCTION

SUCCESSION OF FUNCTIONS

SERIES OF FUNCTIONS

"Exercises of Functions Series"

SIMONE MALACRIDA

In this book exercises are carried out regarding the following mathematical topics:

successions of functions of real variable

series of functions of real variable

diversity of the concepts of convergence of successions and series

Initial theoretical hints are also presented to make the performance of the exercises understood.

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

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ANALYTICAL INDEX

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INTRODUCTION

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I – SUCCESSION OF FUNCTIONS

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

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II – SERIES OF FUNCTIONS

Exercise 1

Exercise 2

Exercises or 3

Exercises or 4

Exercise5

Exercise 6

Exercise7

Exercises or 8

Exercise 9 _

Exercise 10 _

INTRODUCTION

In this exercise book some examples of calculations related to sequences and series of functions of real variable are carried out.

Furthermore, the main concepts of convergence are presented.

Series expansions are a very powerful mathematical tool for describing functions in complementary forms and are widely used in the application fields of physics and technology.

In order to understand in more detail what is explained in the resolution of the exercises, the reference theoretical context is recalled at the beginning of each chapter.

What is exposed in this workbook is generally addressed in advanced mathematical analysis courses (analysis 2) even if the notions present can be easily understood with elements of basic mathematical analysis.

I

SUCCESSION OF FUNCTIONS

A sequence of functions is a sequence whose terms are functions.

Given a sequence of functions and a metric space, it is said to converge pointwise if it happens for every point belonging to the domain:

Define the following sequence:

If it is well defined and the limit of the nth term is zero, the sequence is said to converge uniformly.

If the sequence converges uniformly, the following properties hold: the limit of a sequence of continuous functions is a continuous function just as the limit of differentiable or integrable functions is a derivable or integrable function or the limit of limited or uniformly continuous functions.

Furthermore, the limit of the integrals of a sequence of functions is the integral of the limit just as the limit of the derivatives of a sequence of functions is the derivative of the limit, i.e. it is possible to exchange the signs of limit, sum, integral and derivative if there is uniform convergence.

Dini's lemma states that if a sequence of functions converges punctually and monotonously and the functions are continuous in a compact set, then the sequence converges uniformly.

Given a sequence of functions defined in an open set, it converges both pointwise and uniformly if and only if Cauchy's convergence criterion holds.

Exercise 1

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Determine the punctual limit of the following sequence and establish whether the convergence is uniform.

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The punctual limit of the sequence is calculated as follows:

In the interval considered, therefore, the function tends to the null function.

For uniform convergence, we study the following limit: