MATH20111 - 2008/2009

 

Unit specification

Aims

The programme unit aims to introduce the basic concepts of limit and convergence (of real sequences, series and functions) and to indicate how these are treated rigorously, and then show how these ideas are used in the development of real analysis.

Brief description

The first part of the course discusses the convergence of real sequences and series.

The second part of the course discusses the concept of limit for real-valued functions of a real variable. This concept is then used to define and investigate the concepts of continuity and differentiability for such functions.

Intended learning outcomes

On completion of this unit successful students will be able to:

• evaluate the limits of a wide class of real sequences;
• determine whether or not real series are convergent by comparison with standard series or using the Ratio Test;
• understand the concept of continuity and be familiar with the statements and some proofs of the standard results about continuous real functions;
• understand the concept of the differentiability of a real valued function and be familiar with the statements of the standard results about differentiable real functions.

Future topics requiring this course unit

Real analysis is needed in more advanced courses in analysis, functional analysis and topology and some courses in numerical analysis.

MATH20142 Complex Analysis extends the ideas in this course to functions of a complex variable.

Syllabus

1. Sequences. Null sequences and the standard list of null sequences. Convergent sequences, the Algebra of Limits, divergent sequences, monotone bounded sequences
2. Series. Convergent and divergent series, geometric series and the harmonic series. Series with non-negative terms, the Comparison Test, the Limit Comparison Test, the Ratio Test.
3. Continuity. Limits of real functions, sums, products and quotients of limits. Continuity of real functions, sums, products and quotients of continuous functions, the composition of continuous functions. The standard results about continuous real functions: the Intermediate Value Theorem and the Boundedness Theorem.
4. Differentiability. Differentiability of real-valued functions, sums, products and quotients of differentiable functions, Rolle's Theorem, the Mean Value Theorem, Cauchy's Mean Value Theorem.

Textbooks

• Rod Haggerty, Fundamentals of Mathematical Analysis,Addison-Wesley, second edition 1993.

Learning and teaching processes

Two lectures and one examples class each week.

Assessment

Coursework; Weighting within unit 20%

2 hours end of semester examination; Weighting within unit 80%

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Arrangements