F18CD - Real Analysis

Course leader(s):

James Gaunt (Edinburgh, September)
Nasreddine Megrez (Dubai, September)
Joshua Tan (Malaysia, September)

Aims

The course aims to introduce students to the idea of rigorous mathematical arguments and, in particular, to discuss the rigorous foundations of calculus. An important feature of the course is the use of careful, rigorous proofs of the theorems used and one of the aims of the course is to improve student’s ability to understand such arguments and to develop such proofs for themselves. A central concept in analysis is the idea of convergence, either of sequences, series or of functions, and this course aims to introduce this concept and provide the basic results which will be used in later courses. In addition, it will give methods of obtaining inequalities and approximations (with precise estimates of how good the approximations are), tests for convergence of series and power series and ways of identifying functions defined by power series.

Syllabus

1. Number Systems (1.1 1. Natural numbers, integers, rational numbers and real numbers , 1.2 2. Completeness axiom, 1.3 3. Properties of the real numbers, 1.4 4. Modulus function and its properties, 1.5 5. Suprema and Infima of sets)

2. Sequences (2.1 1. Bounded sequences, 2.2 2. Divergent and convergent sequences, 2.3 3. Properties of convergent sequences, 2.4 4. Monotone sequences and the monotone convergence theorem, 2.5 5. Subsequences and Bolzano-Weierstrass theorem, 2.6 6. Cauchy sequences)

3.1 1. Convergent and divergent series, 3.2 2. Geometric series, 3.3 3. Properties of convergent series, 3.4 4. Divergence test, P-series test, comparison test, alternating series test and ratio test, 3.5 5. Absolute and conditional convergence

4. Functions and Continuity (4.1 1. Bounded functions, 4.2 2. Continuous and discontinuous functions, 4.3 3. Properties of continuous functions, 4.4 4. Functional Limits, 4.5 5. Sequential Continuity, 4.6 6. Intermediate and extreme value theorem)

5. Differentiation and Power Series (5.1 1. Differentiable functions, 5.2 2. Properties of differentiable functions, 5.3 3. Interior, mean value and Rolle's theorem, 5.4 4. Power series, 5.5 5. Radius of convergence, 5.6 6. Taylor series and Lagrange's theorem)

Learning outcomes

By the end of the course, students should be able to do the following:

be able to recall and utilise definitions to prove propositions related to the topics of the real numbers, sequences, series, continuity, differentiability, and power series.
apply the completeness properties of the real numbers to prove the supremum and infimum of given subsets.
apply theorems on sequences and series to assess and prove the convergence of a given sequence or series.
utilise the definitions of continuity of single variable functions to assess and prove the continuity and of a given single variable function.
utilise the definitions of differentiability of single variable functions to assess and prove the differentiability of a given single variable function.
utilise the intermediate value theorem, Rolle’s theorem, the mean value theorem and Fermat’s theorem to prove properties of given continuous and differentiable functions.
to be able to calculate the radius of convergence of a given powers series.
utilise Taylor’s theorem to calculate a power series expression of a given function.

Further details

Curriculum explorer: Click here

SCQF Level: 8

Credits: 15