Convergence of Infinite Sequences and Series - Advanced Calculus (Undergraduate Advanced)

This course provides a complete toolkit for the study of infinite sequences and series. It begins with a rigorous treatment of the convergence of sequences and the algebra of limits. The course then covers the foundational theory of infinite series and provides a comprehensive study of all standard tests for convergence, culminating in an introduction to power series. The concepts of convergence and infinite series are essential in physics, engineering, and computer science. They are used to solve differential equations, analyse signals with Fourier series, calculate probabilities, and determine the accuracy of numerical approximations. This is the mathematical machinery behind precision in scientific modelling. By the end of this course, you will be able to determine if an infinite sequence converges and find its limit. You will also be able to apply the complete suite of convergence tests—including the Integral, Comparison, Ratio, Root, and Alternating Series tests—to determine if an infinite series converges, and find the radius and interval of convergence for any power series. This course is for students who have completed a foundational calculus course. It is the standard curriculum for a second-semester calculus (Calculus II) module and is a direct prerequisite for the study of differential equations, complex analysis, and advanced physics.

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Payment required for enrolment
Enrolment valid for 12 months
This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.
[OAU, Ife] MTH 201: Mathematical Methods I
[OAU, Ife] MTH 201: Mathematical Methods I
This learning track delivers the complete mathematical toolkit required for a university-level science, engineering, or computing degree. It systematically covers the entire MTH 201 curriculum, building from the foundational principles of single-variable calculus - functions, limits, continuity, and differentiability - to the advanced methods of multivariable calculus, infinite series, numerical methods, and ordinary differential equations. This is the definitive preparation for advanced quantitative study. This programme is designed for second-year students offering MTH 201 at Obafemi Awolowo University, Ile-Ife, Nigeria. It is also helpful for any student in a STEM field - including physics, engineering, and computer science - who requires a rigorous and comprehensive command of calculus and its applications. This track delivers a full skill set in mathematical analysis and applied problem-solving. Graduates will be able to solve a wide range of problems, from optimising multivariable functions to modelling dynamic systems with differential equations and testing the convergence of infinite series. This programme directly prepares students for success in advanced courses in vector calculus, partial differential equations, and real analysis, providing the necessary foundation for a career in engineering, data science, or theoretical physics.

This learning track delivers the complete mathematical toolkit required for a university-level science, engineering, or computing degree. It systematically covers the entire MTH 201 curriculum, building from the foundational principles of single-variable calculus - functions, limits, continuity, and differentiability - to the advanced methods of multivariable calculus, infinite series, numerical methods, and ordinary differential equations. This is the definitive preparation for advanced quantitative study. This programme is designed for second-year students offering MTH 201 at Obafemi Awolowo University, Ile-Ife, Nigeria. It is also helpful for any student in a STEM field - including physics, engineering, and computer science - who requires a rigorous and comprehensive command of calculus and its applications. This track delivers a full skill set in mathematical analysis and applied problem-solving. Graduates will be able to solve a wide range of problems, from optimising multivariable functions to modelling dynamic systems with differential equations and testing the convergence of infinite series. This programme directly prepares students for success in advanced courses in vector calculus, partial differential equations, and real analysis, providing the necessary foundation for a career in engineering, data science, or theoretical physics.

Course Chapters

1. Progressions
5
1

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

Chapter lessons

1-1. Definition
12:58

Meaning and types of progressions.

1-2. Arithmetic progressions
27:34

Meaning, examples, and descriptions of arithmetic progressions.

1-3. Geometric progressions
16:00

Meaning, examples, and descriptions of geometric progressions.

1-4. Harmonic progressions
13:33

Meaning, examples, and descriptions of harmonic progressions.

1-5. Sum to infinity
14:37

Sum to infinity of various progressions.

2. Sequences
4
1

Meaning, definitions, and different kinds of sequences.

Chapter lessons

2-1. Meaning
19:01

Meaning of sequences, and how they differ from progressions.

2-2. Defining the terms of a sequence
17:52

Various ways of defining the terms of a sequence.

2-3. Kinds of sequences (1)
20:19

Positive, negative and alternating sequences.

2-4. Kinds of sequences (2)
16:56

Monotone sequences.

3. Convergence of Sequences
1
8

Understanding the convergence of sequences and its proofs.

Chapter lessons

3-1. Definition
25:03

Formal and informal definition of the convergence of a sequence.

4. Limits of Sequences
2
7

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

Chapter lessons

4-1. Theorems (1)
12:04

Some theorems on the limits of real sequences.

4-2. Theorems (2)
30:42

More theorems on the limits of real sequences.

8. Convergence of Alternating Series
3
2

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

Chapter lessons

8-1. Absolute convergence

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

8-2. Conditional convergence

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

8-3. Alternating series test

Test of convergence of alternating series of real numbers.

8. Series of Real Numbers
5
3

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

Chapter lessons

8-1. Definition

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

8-2. Convergence

Meaning of the convergence of a series of real numbers.

8-3. Geometric series

Meaning, conditions for convergence and examples of geometric series.

8-4. Telescoping series

Meaning, conditions for convergence and examples of telescoping series.

8-5. Harmonic series

Divergence of the harmonic series.

9. Theorems on Convergence of Series
4
2

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

Chapter lessons

9-1. A necessary condition

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

9-2. More theorems (1)

More theorems on the convergence of series of real numbers.

9-3. More theorems (2)

More theorems on the convergence of series of real numbers.

9-4. More theorems (3)

More theorems on the convergence of series of real numbers.

9. Power Series
3
3

Definition and convergence of power series.

Chapter lessons

9-1. Definition

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

9-2. Convergence

Examining various states of convergence of power series.

9-3. Radius and interval of convergence

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

10. Convergence of Positive Series
7
5

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

Chapter lessons

10-1. Integral test

Integral test of convergence of a series of real numbers.

10-2. The p-series

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

10-3. Direct comparison test

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

10-4. Limit comparison test

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

10-5. Ratio test

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

10-6. Raabe's test

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

10-7. Nth root test

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