Learning Outcomes:
On completion of this module the student should be able to:
Identify, define, graph (if relevant), and generate examples of the major concepts of sequences and series of real numbers.
Describe and give examples of the relationships between the major concepts of sequences of real numbers.
Verify the veracity of statements about sequences of real numbers and support your answer with a mathematical argument.
State, apply, prove and describe the main theorems relating to sequences of real numbers.
Describe and apply the Axiom of Completeness.
Define and describe the concepts of countable and uncountable sets.
Test a given series for convergence.
Find the interval of convergence of a power series.
Find the Taylor series generated by a given function.
Indicative Module Content:
The following is indicative curricular content. Minor changes may be made throughout the semester.
Section 1. Introduction and Preliminaries
1.1. What is this course about?
1.2 Preliminaries.
Section 2. Sequences - a First Look
2.1. Sequences.
2.2. Null Sequences.
2.3. Convergent Sequences.
Section 3. The Real Numbers - Completeness
3.1. The Axiom of Completeness.
3.2. Consequences of Completeness.
Section 4. Sequences - a More Indepth Look
4.1. The Monotone Convergence Theorem.
4.2. The Bolzano-Weierstrass Theorem.
4.3. Cauchy Sequences and Convergence.
Section 5. Infinite Series
5.1. Infinite Series -- An Introduction.
5.2. Geometric Series.
5.3. Series with Nonnegative Terms.
5.4. Series with Positive and Negative Terms.
Section 6. Power Series
6.1. Introduction.
6.2. Power Series.
6.3. Taylor Series.