II - Convergence: Discrete variables
§1. Convergent sequences and series - §2. Absolutely convergent
series - §3. First concepts of analytic functions
§ 1. Convergent sequences and series
0- Introduction: what is a real number?
Throughout this book, we shall write N for the set of "natural" integers, i.e.
those ~ 0; Z for that of the "rational" integers, i.e. of any sign; Q for the set
ofrational numbers (quotients of two integers); lR. for the set of real numbers;
and C for the set of complex numbers x + iy with x, y E lR. and i 2 = -1: of
course NeZ c Q c lR. c Co
Since we hope that the reader is relatively familiar with N, Z and Q, we
shall emphasise the construction of real numbers, though without providing
all the details, then briefly recall that of C, which is much simpler.
The so-called "real" numbers - it is too late to change the terminology -
are truly not to be met in physical reality; they were born in the brains of
mathematicians. The event which precipitated this process was the discovery
by the Pythagoreans, in the V th century before our era, of the fact that the
ratio ! (1 + V5) between the diagonal and the side of a regular pentagon -
their emblem - and, later, the numbers y'2, y'3, etc. are not rational; if for
example one had y'2 = p/q where p and q are two integers and not both even
(if not, simplify!), then the relation p2 = 2q2 shows that p is even, and so 4
divides the right hand side, whence q is even: contradiction, absolute horror!
For mathematicians.
The Greeks of this period, like their Babylonian, Indian or Egyptian predecessors, only knew of fractions: the successors of the Pythagoreans, until
Euclid, were forced to develop a very abstruse theory of the (positive) real
numbers, grounded in the "measure of magnitudes": to say that the number l
1 The notation 1r was introduced unsuccessfully in 1706 by an Englishman, and
independently by Euler in 1739; since everyone read him, the usage spread. For
§1. Convergent sequences and series - §2. Absolutely convergent
series - §3. First concepts of analytic functions
§ 1. Convergent sequences and series
0- Introduction: what is a real number?
Throughout this book, we shall write N for the set of "natural" integers, i.e.
those ~ 0; Z for that of the "rational" integers, i.e. of any sign; Q for the set
ofrational numbers (quotients of two integers); lR. for the set of real numbers;
and C for the set of complex numbers x + iy with x, y E lR. and i 2 = -1: of
course NeZ c Q c lR. c Co
Since we hope that the reader is relatively familiar with N, Z and Q, we
shall emphasise the construction of real numbers, though without providing
all the details, then briefly recall that of C, which is much simpler.
The so-called "real" numbers - it is too late to change the terminology -
are truly not to be met in physical reality; they were born in the brains of
mathematicians. The event which precipitated this process was the discovery
by the Pythagoreans, in the V th century before our era, of the fact that the
ratio ! (1 + V5) between the diagonal and the side of a regular pentagon -
their emblem - and, later, the numbers y'2, y'3, etc. are not rational; if for
example one had y'2 = p/q where p and q are two integers and not both even
(if not, simplify!), then the relation p2 = 2q2 shows that p is even, and so 4
divides the right hand side, whence q is even: contradiction, absolute horror!
For mathematicians.
The Greeks of this period, like their Babylonian, Indian or Egyptian predecessors, only knew of fractions: the successors of the Pythagoreans, until
Euclid, were forced to develop a very abstruse theory of the (positive) real
numbers, grounded in the "measure of magnitudes": to say that the number l
1 The notation 1r was introduced unsuccessfully in 1706 by an Englishman, and
independently by Euler in 1739; since everyone read him, the usage spread. For