Table of Contents
Table of Contents
“Exercises of Numerical Series”
INTRODUCTION
THEORY OF NUMERICAL SERIES
EXERCISES
“Exercises of Numerical Series”
“Exercises of Numerical Series”
SIMONE MALACRIDA
In this book exercises are carried out regarding the following mathematical topics:
numerical series
convergence criteria
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.
ANALYTICAL INDEX
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INTRODUCTION
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I – THEORY OF NUMERICAL SERIES
Definitions
Operations
Notable series
Convergence criteria
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II – EXERCISES
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 1 4
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
Exercise 20
Exercise 21
Exercise 22
Exercise 23
Exercise 24
Exercise 25
Exercise 26
Exercise 27
Exercise 28
Exercise 29
Exercise 30
Exercise 31
Exercise 32
INTRODUCTION
INTRODUCTION
In this exercise book, some examples of calculations relating to numerical series and the respective convergence criteria are carried out.
Understanding the fundamentals of numerical series is fundamental for tackling the subsequent steps, such as the one represented by the series of functions or by their series expansions.
Furthermore, numerical series are well suited for a mathematical generalization of practical and applicative problems, already known since ancient times.
In order to understand in more detail what is presented in the resolution of the exercises, the theoretical reference context is recalled in the first chapter.
Although no background in mathematical analysis is necessary to understand numerical series (and therefore their study can also be undertaken at the upper secondary level), the different concepts of convergence presuppose, at least, a topological notion that is only an introduction to analysis can support.
I
THEORY OF NUMERICAL SERIES
THEORY OF NUMERICAL SERIES
Definitions
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A series is the sum of the elements of a sequence and is a generalization of the addition operation, extended to the case of infinitely many terms.
The series symbol is called summation and is indicated as follows:
while the extreme variations of the sum are indicated at the bottom and top:
If the upper index is finite we speak of a sequence of partial sums, if it is infinite we speak of a true and proper series.
The series is said to be convergent if there is a finite number which equals the sum of all its elements.
This value is called the sum of the series.
If this limit is infinite, the series is divergent, if instead the limit does not exist, the series is called oscillating .
Convergent and divergent series are called regular .
A necessary condition of convergence of a series is that the infinite nth element is zero.
A numerical series is a series whose elements are numbers.
A series is said to have positive terms when all its terms are positive real numbers.
A numerical series converges if and only the Cauchy convergence criterion is satisfied :
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Operations
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The sum of the series equals the series of
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 20.04.2023
ISBN: 978-3-7554-3967-7
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