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.

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.
MTH 207: Real Analysis I
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.

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.

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Course Chapters

1. Progressions
5
1
Meaning of progressions; review of arithmetic, geometric, harmonic progressions and their sum to infinity.
Concept Overviews
5 Lessons
1:24:42
Problem Walkthroughs
1 Lesson
16:24
2. Sequences
4
1
Meaning, definitions, and different kinds of sequences.
Concept Overviews
4 Lessons
1:14:08
Problem Walkthroughs
1 Lesson
19:59
3. Convergence of Sequences
1
8
Understanding the convergence of sequences and its proofs.
Concept Overviews
1 Lesson
25:03
Problem Walkthroughs
8 Lessons
3:43:03
4. Limits of Sequences
2
7
Theorems on limits of sequences and evaluation of limits of sequences.
Concept Overviews
2 Lessons
42:46
Problem Walkthroughs
7 Lessons
2:41:06
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.
Concept Overviews
3 Lessons
Problem Walkthroughs
2 Lessons
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.
Concept Overviews
5 Lessons
Problem Walkthroughs
3 Lessons
9. Theorems on Convergence of Series
4
2
Examining some theorems on the convergence and divergence of series of real numbers.
Concept Overviews
4 Lessons
Problem Walkthroughs
2 Lessons
9. Power Series
3
3
Definition and convergence of power series.
Concept Overviews
3 Lessons
Problem Walkthroughs
3 Lessons
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.
Concept Overviews
7 Lessons
Problem Walkthroughs
5 Lessons