Analysis of Infinite Sequences and Series - Real Analysis (Undergraduate Advanced)

This course provides the formal, proof-based foundation for the concepts of convergence covered in calculus. It moves beyond computation to explore the deep theoretical questions of why sequences and series converge. The material covers the topological properties of sequences, key convergence theorems like Bolzano-Weierstrass, Cauchy sequences, and the advanced topic of uniform convergence for series of functions. This subject is the gateway to higher mathematics. The rigorous proof-writing and analytical skills developed here are non-negotiable for any student pursuing a degree in pure or applied mathematics. These principles are the theoretical bedrock upon which advanced concepts in differential equations, functional analysis, and theoretical physics are built. By the end of this course, you will be able to construct rigorous proofs for statements about the convergence of sequences. You will also be able to apply the Monotone Convergence and Bolzano-Weierstrass theorems, use the Cauchy Criterion to prove convergence, and analyse the uniform convergence of a series of functions using the Weierstrass M-Test. This course is for mathematics majors and advanced students who have already completed a full Calculus II course. It is the standard curriculum for a first module in Real Analysis and is an essential prerequisite for graduate-level study in mathematics, physics, and theoretical computer science.

Enrolment valid for 12 months

This course is also part of the following learning tracks. You may join a track to gain comprehensive knowledge across related courses.

MTH 207: Real Analysis I

Real Analysis bridges the critical gap between computational calculus and rigorous advanced mathematics. This learning track delivers the complete NUC CCMAS MTH 207 curriculum, transitioning you from intuitive understanding to formal mathematical proof. It establishes the theoretical foundation required for serious modelling in science, engineering, and pure mathematics. This programme is targeted at mathematics majors and advanced undergraduates in physics and engineering who have completed foundational calculus. It is designed for students requiring the rigorous analytical skills demanded by graduate-level studies and theoretical research. You will master the construction of rigorous proofs for sequence and series convergence, applying cornerstone theorems like Bolzano-Weierstrass and Cauchy criteria. You will achieve a formal command of continuity and differentiability, deriving major calculus rules from first principles. Completion provides the non-negotiable prerequisite knowledge for advanced studies in functional analysis, differential equations, and theoretical physics.

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.

Course Chapters

1. Introduction

This chapter introduces the field of real analysis. It explains the purpose of the subject, which is to provide the rigorous, proof-based foundation for the concepts of calculus, moving the focus from 'how' to 'why'. Key learning objectives include understanding the need for mathematical rigour and appreciating the role of formal proof in establishing the certainty of mathematical truths.

Concept Overviews

2 Lessons

2. Boundedness of Sequences

This chapter covers the foundational properties of sequences of real numbers. It details the concepts of boundedness, supremum, and infimum, which are essential for proving the major convergence theorems. Key learning objectives include: determining if a sequence is bounded; finding the supremum and infimum of a set; and understanding the formal definition of a subsequence.

Concept Overviews

3 Lessons

55:16

Problem Walkthroughs

3 Lessons

47:16

3. Theorems on Convergence of Sequences

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

Concept Overviews

5 Lessons

Problem Walkthroughs

2 Lessons

4. Sequences of Functions

Definition and convergence of sequences of real-valued functions.

Concept Overviews

3 Lessons

Problem Walkthroughs

3 Lessons

5. Series of Functions

Definition and convergence of series of real-valued functions.

Concept Overviews

4 Lessons

Problem Walkthroughs

2 Lessons