1. -1-
A NONSTANDARD STUDY
ON THE
TAYLOR SERIES DEVELOPMENT
A THESIS
SUBMITTED TO THE COLLEGE OF SCIENCE
UNIVERSITY OF SALAHADDIN-ERBIL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
IN MATHEMATICS
BY
IBRAHIM OTHMAN HAMAD
B.Sc. (MATHEMATICS)-1992
JUNE JOZARDAN
2000 2700

2. -3-
Abstract
The main aim of the present work is to use some concepts of
nonstandard analysis given by Robinson, A. [26] and axiomatized by
Nelson, E.[23] to associate to the classical formula of the remainder
majoration, that plays an incontestable role, with an approximation of this
remainder. More precisely, we try to find a connection between the
remainder and the development of the next nonzero term by defining
)(
)(
)( 1
xT
xR
x
n
n
n
−
=Φ as an approximation factor of )(xTn w.r.t )(xf for a
Taylor series ∑
∞
=0
)(
k
n xT . Moreover under certain conditions we shall prove
the following:
i) The approximation factor 1)( ≅Φ xn for limited n.
ii) The approximation factor ∑
∞
=
≅Φ
0
)(
n
n
nn zcx for unlimited index inside the
convergence disc where )(
ω
+ω
=
a
a
c n
n
o
.
iii) The approximation factor ∑
∞
=
−≅Φ
1
)(
n
n
n
n
z
d
x for unlimited index outside
the convergence disc, where )(
ω
−ω
=
a
a
d n
n
o
.
We also consider the analyticity of the approximation factor. It is
proved that the approximation factor ∑=Ψ n
n
n
u
u
c
u)(o
where )(
ω
+ω
=
a
a
c n
n
o
,
and
)0(
)(
)( )(
)(
ω
ω
ω
ω=Ψ
f
u
f
u .

3. -4-
Contents
Page No.
List of Symbols 10
Introduction 11
Chapter One Basic Concepts
1-1 Backgrounds and Definitions 16
1-2 An Asymptotic Approximation of Series 23
Chapter Two A Nonstandard Approximation Using
Taylor Series
2-1 Introduction 29
2-2 A Shadow Determination of an Approximation Factor
for Standard n and 0≅ε . 33
2-3 A Shadow Determination of an Approximation Factor
for Unlimited Index Inside the Convergence Disc. 34
2-4 A Shadow Determination of an Approximation Factor
for Unlimited Index Outside the Convergence Disc. 40
2-5 Motivation of a General Study of an Approximation
Factor for Unlimited Index. 44
Chapter Three Analyticity of the Approximation Factor
3-1 Analyticity of the Approximation Factor (Special Case) 47
3-2 Analyticity of the Approximation Factor (General Case) 51
References 57

4. -5-
Table of Symbols
Symbols Description
Infinitely Close
)(≅< Less than but not Infinitely Close
st
∀ For All Standard
fin
∀ For All Finite
finst
∀ For All Standard Finite
)(xnΦ Approximation Factor of Order n of x
)(xTn Term of Degree n as a function of x
xo
Shadow of x
)(xm Monad of x
)(xgal Galaxy of x
N Set of Standard Natural Numbers
N Set of unlimited Natural Numbers
εγβα ,,, Infinitesimal Numbers
ω Infinitely Large Number
∞
C Space of Every Where Differentiable Functions

5. -6-
Introduction
The primary idea about nonstandard analysis goes back to the problem of
determining the slope of a tangent of a curve, limits and derivatives in which
Newton, I(1642-1727) and Leibniz(1646-1716) worked out to approximate the
tangent of a curve by a line which intersects the curve at two points such that the
distance between them is infinitely small. Such distance, was named by Newton
small quantity (or infinitesimal)[4] [9] [29].
Many attempts had been done to establish the foundations of infinitesimals.
Leibniz and his followers were never been able to state with sufficient precision
just what rules were supposed to govern their new system including ‎infinitely ‎
small as well as infinitely large quantities with no contradiction in these rules, until
Abraham Robinson in (1961), presented a complete and satisfactory solution of
Leibniz’s problem by formulating the ideal quantities (i.e., infinitely small and
‎infinitely large) in precise mathematical structures under the name (Nonstandard
Analysis). More precisely, Robinson showed that there exists a proper extension,
say *R, of the field of real numbers R which in a certain sense have the same
formal properties as R and is non-Archimedean [4] [26] [29].
Later on some mathematicians tried to reconstruct nonstandard analysis
models using the set theory [13] [23] [29], such as Luxembourg (1962), and
Nelson, E. (1977) who presented a great and illustrative construction using the
axiomatic set theory of ZFC (Zermelo-Fraenckel Set Theory with Axiom of
Choice)[31] . Today, the nonstandard analysis regarded as a technique rather
than a subject. There are problems, the mathematicians were unable to prove
them using conventional methods, while they can be proved using nonstandard
methods, such as Bernstein Robinson theorem [4].
In the present work, we use some concepts of nonstandard analysis that
are axiomatized by Nelson, E. [23]. In practice, an approximation is successful if
the error is small. For an approximation study, it is, therefore interesting to

6. -7-
arrange a mathematical theory, which permits us to express simply the notion”
be small” [15] [24].
The nonstandard analysis makes this possible, especially the axiomatic
version IST (Internal Set Theory) proposed by Nelson, E. It happens, that one
side of the extension of set theory’s language ZFC is by introducing predicate
“standard”, while the other side is by adding to ZFC some axioms concerning
the usage of the new predicate. Two of the principle consequences of these
axioms are:
1- each set defined in ZFC is standard.
2- each infinite set defined in ZFC pssesses nonstandard elements.
Any collection of real numbers with predicates (standard, infinitesimal,
limited, unlimited,…ect)is not a set in the axiomatic sense of ZFC , and it is
called an external set.
Now we point out three interferences of the external sets to our study of
approximations.
Firstly, for a given accuracy, we can form the “ external” set of numbers
for what the approximation attains that accuracy. We have used this possibility
in chapter two to describe sketches, by using a computer program written in a
Visual Basic Language Version 5, showing the approximation of a function by
it’s Taylor polynomials. We have characterized the collection of points for
which the graphs of the functions and their Taylor polynomials are conform to
the naked eye, by an external set of points such that the remainder is
infinitesimal.
Secondly, the external sets interfere to a practical level for reasoning
concerning the approximation. They are at the base of “ permanence principles”
of set statements, which enable us to understand the validity of a proposition
beyond the domain where it is proved to be effective.
Thirdly, the external sets can be used to distinguish certain speeds of the
approximation considered as the approximation of a standard real number “a ”

7. -8-
by the terms of the standard sequence Nnnu ∈}{ which converges to this number.
For example, ea = and ∑
=
=
n
k
n
k
u
0 !
1
. This can be expressed by the sentence “ for
every non limited ∈n N, nu belongs to a monad of “a ”. This sentence connects
the two external sets of unlimited positive integers and that of real numbers at an
infinitesimal distance from “a “.
In the theory of asymptotic development, we consider particularly the
functions f that are approximated by a sequence of functions NnnP ∈}{ such
that the remainder nn PfxR −=)( , satisfies the property )()( n
n xxR ο= , if
0→x , for every ∈n N. This property is in particular, satisfied in the case where
f in C
∞
, and nP its Taylor polynomial. Moreover, notice that if f is standard,
then the proposition “ monadR n
n −∈εε)( for every standard n , and 0≅ε , is
satisfied. This constitutes a connection point between the classical method and our
nonstandard method [2] [22] [25] [30].
Another correspondence is situated at the terminology level that can be
illustrated by the following example: consider a problem of the behavior of a
family of functions depending on a parameter “a ”, at infinity. It is then of a
classical use to distinguish between “numbers depending on x ” ( )(xaa = ),
susceptible to increase over every value, and also can influence the asymptotic
behavior and a “fixed number”, independent of x , and have no influence on the
asymptotic behavior. We compare this with a nonstandard distinction between
“numbers is depending on ω,ωunlimited positive” ( )(ωaa = ) and “standard
numbers”. But however, we observe an important difference. The first distinction
does not permit us to separate the two types of numbers on the real line, while the
second distinction makes it possible. For unlimited numbers )(ωa , which are
greater than the standard numbers, but a number )(xa “depends on x ” is always
going beyond the fixed numbers.

8. -9-
In this thesis, which consists of three chapters we tried to study a
nonstandard approximation, using Taylor polynomials. We have tried to make this
work a self-contained as much as possible.
Chapter one was written in order to provide the reader with general
background, notions and materials needed. It consists of two sections, the first
gives a brief description of IST and other concepts and terminologies of
nonstandard analysis needed. The second section contains some theorems and
results concerning (nonstandard approximation of series).
Chapters two and three considered to be the climax of the work of this
thesis. Chapter two consists of fife sections, in which we study the
approximation of a function using Taylor polynomials. In the first section, we
discuss the classical notion about the remainder and explaining its
disadvantages. In the second section we study the shadow of the approximation
factor for Taylor Series for standard n and 0≅ε . In the sections three and
fourth, we tried to determine the shadow of the approximation factor for Taylor
series inside and outside the disc of convergence. In the fifth section, we give a
motivation of a general study of the approximation factor. In chapter three, we
study the analyticity of the approximation factor in two cases; special and
general.