Convergence of Infinite Sequences and Series - Advanced Calculus

Do you want to learn how to work with infinite sums of numbers and functions and their properties, operations, and applications? Do you want to understand the concepts of progressions, sequences, series, and power series, and how they relate to the approximation, convergence, and divergence of functions? Do you want to master the skills of finding and applying various tests and criteria of convergence and divergence to different types of series, such as arithmetic, geometric, harmonic, p-series, and alternating series? If you answered yes to any of these questions, then this course is for you! This course reviews the fundamental concepts of finite progressions and provides a thorough treatise of the convergence of infinite real sequences and series. You will learn how to: - Define and classify progressions and their properties, such as common difference, common ratio, and sum to infinity - Define and classify sequences and their properties, such as terms, general term, and boundedness - Find the convergence or divergence of a sequence using the formal definition and various theorems and examples - Define and classify series and their properties, such as partial sums, absolute and conditional convergence, and uniform convergence - Find the convergence or divergence of a series of real numbers using various methods and techniques, such as direct comparison, integral test, p-series test, ratio test, Raabe's test, and alternating series test - Define and classify sequences and series of functions and their properties, such as pointwise and uniform convergence, and term-by-term differentiation and integration - Find the convergence or divergence of a sequence or series of functions using various methods and techniques, such as Weierstrass M-test, Abel's test, and Dirichlet's test - Define and classify power series and their properties, such as radius and interval of convergence, and term-by-term differentiation and integration - Find the convergence or divergence of a power series using various methods and techniques, such as ratio test, root test, and Cauchy-Hadamard theorem This course is suitable for anyone who wants to learn or review the advanced topics of calculus and its applications. It is especially useful for students and professionals in analysis, differential equations, optimization, physics, engineering, and other related fields. By the end of this course, you will have a solid foundation of the theory and practice of calculus and its operations. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve infinite sums of numbers and functions and their approximation, convergence, and divergence. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

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This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.
MTH 201: Mathematical Methods I
MTH 201: Mathematical Methods I
Comprehensive treatise of advanced calculus covering limits, continuity and differentiability, infinite sequences and series, partial differentiation, numerical methods and ordinary differential equations. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Comprehensive treatise of advanced calculus covering limits, continuity and differentiability, infinite sequences and series, partial differentiation, numerical methods and ordinary differential equations. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Course Chapters

1
Progressions

Meaning of progressions; review of arithmetic, geometric, harmonic progressions and their sum to infinity.

Chapter lessons

1.Definition12:58

Meaning and types of progressions.

2.Arithmetic progressions27:34

Meaning, examples, and descriptions of arithmetic progressions.

3.Geometric progressions16:00

Meaning, examples, and descriptions of geometric progressions.

4.Harmonic progressions13:33

Meaning, examples, and descriptions of harmonic progressions.

5.Sum to infinity14:37

Sum to infinity of various progressions.

6.Worked examples16:24

Worked examples on sum of the terms of a progression.

2
Sequences

Meaning, definitions, and different kinds of sequences.

Chapter lessons

1.Meaning19:01

Meaning of sequences, and how they differ from progressions.

2.Defining the terms of a sequence17:52

Various ways of defining the terms of a sequence.

3.Kinds of sequences (1)20:19

Positive, negative and alternating sequences.

4.Kinds of sequences (2)16:56

Monotone sequences.

5.Worked examples19:59

Worked examples on identifying different kinds of sequences.

3
Convergence of Sequences

Understanding the convergence of sequences and its proofs.

Chapter lessons

1.Definition25:03

Formal and informal definition of the convergence of a sequence.

2.Worked examples (1)39:33

Worked examples on proving the convergence of real sequences.

3.Worked examples (2)35:45

More worked examples on proving the convergence of real sequences.

4.Worked examples (3)17:32

More worked examples on proving the convergence of real sequences.

5.Worked examples (4)22:29

More worked examples on proving the convergence of real sequences.

6.Worked examples (5)31:50

More worked examples on proving the convergence of real sequences.

7.Worked examples (6)18:37

More worked examples on proving the convergence of real sequences.

8.Worked examples (7)21:58

More worked examples on proving the convergence of real sequences.

9.Worked examples (8)35:19

More worked examples on proving the convergence of real sequences.

4
Limits of Sequences

Theorems on limits of sequences and evaluation of limits of sequences.

Chapter lessons

1.Theorems (1)12:04

Some theorems on the limits of real sequences.

2.Theorems (2)30:42

More theorems on the limits of real sequences.

3.Worked examples (1)16:27

Worked examples on evaluating limits of real sequences.

4.Worked examples (2)16:59

More worked examples on evaluating limits of real sequences.

5.Worked examples (3)27:10

More worked examples on evaluating limits of real sequences.

6.Worked examples (4)25:22

More worked examples on evaluating limits of real sequences.

7.Worked examples (5)22:49

More worked examples on evaluating limits of real sequences.

8.Worked examples (6)19:09

More worked examples on evaluating limits of real sequences.

9.Worked examples (7)33:10

More worked examples on evaluating limits of real sequences.

5
Boundedness of Sequences

Boundedness of subsets of real numbers, upper and lower bounds, boundedness of sequences, and related matters.

Chapter lessons

1.Definition28:53

Meaning of boundedness of a set of real numbers.

2.Infimum and supremum (1)10:32

Meaning of infimum and supremum of subsets of real numbers.

3.Infimum and supremum (2)15:51

Properties of infimum and supremum of subsets of real numbers.

4.Worked examples (1)17:33

Worked examples on boundedness of subsets of real numbers.

5.Worked examples (2)9:35

More worked examples on boundedness of subsets of real numbers.

6.Worked examples (3)20:08

More worked examples on boundedness of subsets of real numbers.

6
Theorems on Convergence of Sequences

Theorems and criteria for convergence of sequences of real numbers - monotone convergence theorems, Bolzano-Weierstrass theorem, Cauchy criterion, etc.

Chapter lessons

1.Monotone convergence

Theorems on convergence of monotone sequences of real numbers.

2.Subsequences

Definition and examples of subsequences of real numbers.

3.Bolzano-Weierstrass theorem

The Bolzano-Weierstrass theorem on convergence of subsequences of real numbers.

4.Cauchy sequence

Formal and informal definitions of the Cauchy sequence with examples.

5.Cauchy convergence criterion

The Cauchy convergence criterion for sequences of real numbers and its implications.

6.Worked examples (1)

Worked examples on establishing the convergence of sequences of real numbers using convergence theorems.

7.Worked examples (2)

More worked examples on establishing the convergence of sequences of real numbers using convergence theorems.

7
Sequences of Functions

Definition and convergence of sequences of real-valued functions.

Chapter lessons

1.Definition

Meaning and examples of sequences of functions.

2.Pointwise convergence

Definition of pointwise convergence of sequence of functions.

3.Uniform convergence

Definition of uniform convergence of sequences of functions.

4.Worked examples (1)

Worked examples on pointwise and uniform convergence of sequences of functions.

5.Worked examples (2)

More worked examples on pointwise and uniform convergence of sequences of functions.

6.Worked examples (3)

More worked examples on pointwise and uniform convergence of sequences of functions.

8
Series of Real Numbers

Meaning and convergence of series of real numbers; evaluating the convergence (sum to infinity) of some special series.

Chapter lessons

1.Definition

Meaning of a series of real numbers, and how it differs from progressions and sequences.

2.Convergence

Meaning of the convergence of a series of real numbers.

3.Geometric series

Meaning, conditions for convergence and examples of geometric series.

4.Telescoping series

Meaning, conditions for convergence and examples of telescoping series.

5.Harmonic series

Divergence of the harmonic series.

6.Worked examples (1)

Worked examples on evaluating some real series.

7.Worked examples (2)

More worked examples on evaluating some real series.

8.Worked examples (3)

More worked examples on evaluating some real series.

9
Theorems on Convergence of Series

Examining some theorems on the convergence and divergence of series of real numbers.

Chapter lessons

1.A necessary condition

Examination of a necessary but not sufficient condition for the convergence of a series of real numbers.

2.More theorems (1)

More theorems on the convergence of series of real numbers.

3.More theorems (2)

More theorems on the convergence of series of real numbers.

4.More theorems (3)

More theorems on the convergence of series of real numbers.

5.Worked examples (1)

Worked examples on establishing the convergence or divergence of series of real numbers using theorems on convergence.

6.Worked examples (2)

More worked examples on establishing the convergence or divergence of series of real numbers using theorems on convergence.

10
Convergence of Positive Series

Tests of convergence of series of positive (or ultimately-positive) real numbers - integral, comparison, ratio, Raabe's, etc. tests.

Chapter lessons

1.Integral test

Integral test of convergence of a series of real numbers.

2.The p-series

Establishing the condition for convergence of the p-series using the integral test.

3.Worked examples (1)

More worked examples on the integral test of convergence of a series of real numbers.

4.Direct comparison test

Direct comparison test of convergence of a series of real numbers.

5.Limit comparison test

Limit comparison test of convergence of a series of real numbers.

6.Worked examples (3)

Worked examples on comparison tests of convergence of a series of real numbers.

7.Worked examples (4)

More worked examples on comparison tests of convergence of a series of real numbers.

8.Ratio test

The ratio test of convergence of a series of real numbers.

9.Raabe's test

Raabe's test of convergence of a series of real numbers.

10.nth root test

The nth root test of convergence of a series of real numbers.

11.Worked examples (5)

More worked examples on tests of convergence of series of real numbers.

12.Worked examples (6)

More worked examples on tests of convergence of series of real numbers.

11
Convergence of Alternating Series

Tests of convergence of series of arbitrary (positive or negative) real numbers; conditional and absolute convergence of alternating series of real numbers.

Chapter lessons

1.Absolute convergence

Meaning and implication of absolute convergence for an alternating series of real numbers.

2.Conditional convergence

Meaning and implication of conditional convergence for an alternating series of real numbers.

3.Alternating series test

Test of convergence of alternating series of real numbers.

4.Worked examples (1)

Worked examples on the condition of convergence of a series of real numbers.

5.Worked examples (2)

More worked examples on the convergence of alternating series of real numbers.

12
Series of Functions

Definition and convergence of series of real-valued functions.

Chapter lessons

1.Definition

Meaning and examples of series of functions.

2.Convergence

Meaning of convergence for series of functions.

3.Cauchy criterion

Cauchy criterion for convergence of series of functions.

4.Weierstrass M-test

Weierstrass M-test for convergence of series of functions.

5.Worked examples (1)

Worked examples on convergence of series of functions.

6.Worked examples (2)

More worked examples on convergence of series of functions.

13
Power Series

Definition and convergence of power series.

Chapter lessons

1.Definition

Meaning and examples of power series; centre of convergence of power series.

2.Convergence

Examining various states of convergence of power series.

3.Radius and interval of convergence

Meaning and expressions for radius and interval of convergence of power series.

4.Worked examples (1)

Worked examples on centre, radius and interval of convergence of power series.

5.Worked examples (2)

More worked examples on centre, radius and interval of convergence of power series.

6.Worked examples (3)

More worked examples on centre, radius and interval of convergence of power series.