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Analysis I (part 3): Sequences of real numbers I and II

École Polytechnique FĂ©dĂ©rale de Lausanne via EdX

Description

A sequence of real numbers is a function f:N→R. It is usual to write an:=f(n) for the value of f in n. For example, we could define a sequence f(n):=an:=12n, that is to say a0=1,a1=12,a2=14,a3=18,... . The central concept is that of the limit of a sequence: it is a real number to which, intuitively, the given sequence approaches more and more. For example, the sequence an given above admits the number zero as a limit. We will define the concept of the limit in a rigorous manner and develop methods to establish the existence of a limit. In addition, we will discover a link between the concept of the limit and that of the infimum and the supremum of a set. A very important application of sequences of real numbers is the fact that each real number can be considered as the limit of a sequence of rational numbers. We will see how to obtain the irrational number ratio of 5 as the limit of a sequence of rational numbers. We study the concept of Cauchy sequences and sequences defined by linear induction. We show certain properties of sequences defined by linear induction, making a connection with Cauchy sequences. We are interested in the limits of sequences and subsequences, which brings us to the Bolzano-Weierstrass theorem. Using sequences, we also define the concept of numerical series which we illustrate using different examples. We define certain convergence criteria for the series, notably the d'Alembert criterion, the Cauchy criterion, the comparison criterion and the Leibniz criterion. Finally, we study numerical series with one parameter.

Tags
Economics

Syllabus

Chapter 3: Sequences of real numbers, I

3.1 Definitions and examples

3.2 Sequences defined by induction

3.3 Basic properties

3.4 Limit of a sequence

3.5 Two proposals

3.6 Divergent sequences

3.7 Algebraic operations on limits

3.8 Theorem of the two policemen

3.9 Monotone sequences

3.10 Convergence of a sequence defined by induction

3.11 Good to know

Chapter 4: Sequences of real numbers, II

1.1 Rational numbers, properties

1.2 Axiomatic introduction of R

1.3 Minimum

1.4 Supreme

1.5 Real numbers, sqrt(2)

1.6 Subsets of R

1.7 Absolute value

1.8 Additional properties of R

Analysis I (part 3): Sequences of real numbers I and II

École Polytechnique FĂ©dĂ©rale de Lausanne
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A sequence of real numbers is a function f:N→R. It is usual to write an:=f(n) for the value of f in n. For example, we could define a sequence f(n):=an:=12n, that is to say a0=1,a1=12,a2=14,a3=18,... . The central concept is that of the limit of a sequence: it is a real number to which, intuitively, the given sequence approaches more and more. For example, the sequence an given above admits the number zero as a limit. We will define the concept of the limit in a rigorous manner and develop methods to establish the existence of a limit. In addition, we will discover a link between the concept of the limit and that of the infimum and the supremum of a set. A very important application of sequences of real numbers is the fact that each real number can be considered as the limit of a sequence of rational numbers. We will see how to obtain the irrational number ratio of 5 as the limit of a sequence of rational numbers. We study the concept of Cauchy sequences and sequences defined by linear induction. We show certain properties of sequences defined by linear induction, making a connection with Cauchy sequences. We are interested in the limits of sequences and subsequences, which brings us to the Bolzano-Weierstrass theorem. Using sequences, we also define the concept of numerical series which we illustrate using different examples. We define certain convergence criteria for the series, notably the d'Alembert criterion, the Cauchy criterion, the comparison criterion and the Leibniz criterion. Finally, we study numerical series with one parameter.

Chapter 3: Sequences of real numbers, I

3.1 Definitions and examples

3.2 Sequences defined by induction

3.3 Basic properties

3.4 Limit of a sequence

3.5 Two proposals

3.6 Divergent sequences

3.7 Algebraic operations on limits

3.8 Theorem of the two policemen

3.9 Monotone sequences

3.10 Convergence of a sequence defined by induction

3.11 Good to know

Chapter 4: Sequences of real numbers, II

1.1 Rational numbers, properties

1.2 Axiomatic introduction of R

1.3 Minimum

1.4 Supreme

1.5 Real numbers, sqrt(2)

1.6 Subsets of R

1.7 Absolute value

1.8 Additional properties of R

Tags

Economics

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