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MC248 Further Real Analysis


MC248 Further Real Analysis

Credits: 10 Convenor: Dr C. Eaton Semester: 1 (weeks 1-6)


Prerequisites: essential: MC146
Assessment: Individual and group coursework: 20% One and a half hour hour exam: 80%

Lectures: 18 Classes: 5
Tutorials: none Private Study: 52
Labs: none Seminars: none
Project: none Other: none
Total: 75


Explanation of Pre-requisites

This module continues the study, started in MC146, of real analysis and its applications; it thus builds directly upon the material studied in MC146.

Course Description

The trigonometric functions $\sin x$, $\cos x$, $\tan x$ and the exponential function ex permeate the whole of mathematics. Indeed, you've undoubtedly used them extensively, and feel comfortable with them. But do you know what they are? or more quantitatively, can you calculate their value for a given x to, say, an accuracy of 30 significant figures? You can probably give a definition for the trigonometric functions in terms of ratios of sides of a right-angled triangle, but this is no good for calculation purposes. (Try determining $\sin \pi/4$ to 3 significant figures by drawing an appropriate triangle and measuring its sides with a ruler.) And you can try to explain that ex is the number e multiplied by itself x times. But what if x is not an integer? And what is e anyway?

The key to rigourising our understanding of such functions is through the study of infinite series: these are the fundamental objects studied in the course. Now an infinite series (a sum of infinitely many numbers) is a questionable object--we can only sum things together one at a time, so how can we hope to make anything meaningful from the sum of infinitely many quantities. This module explains how some infinite series can be considered to represent bona fide quantities--such series are called convergent, and how such series can be distinguished from those that are divergent. It then becomes possible to define functions, in particular the trigonometric and exponential functions, as infinite series. And having done so, it is natural to want to differentiate and integrate such functions. But (and haven't we heard this before) what is really meant by differentiation and integration? The module also puts these topics on a firm footing with rigorous definitions based on the notions of a limit (remember MC146) and of, guess what, infinite series.

Objectives

To know the definitions of, and understand the key concepts introduced in, this module.

To understand, reconstruct and apply the main results and proofs covered in the module.

To know the definition of convergence for infinite series, and test for convergence using standard tests.

To know the formal definitions of differentiation and Riemann integration.

Transferable Skills

Basic mathematical analysis. Groupwork.

Syllabus

Sequences:

Subsequences, Bolzano-Weierstrass Theorem, Cauchy sequences.


Series:

Infinite series, geometric series, harmonic series, comparison test, ratio test, alternating series, conditional convergence, sums and products of series, power series, radius of convergence.


Differentiation:

Local maximum and minimum, Rolle's theorem, Mean Value Theorem, differentiation of power series, Taylor's series and theorem.


Integration:

Dissection of an interval, upper and lower sums, refinement, upper and lower integrals, integrable functions, fundamental theorem of calculus, limitations of the theorem of Riemann integration.

Reading list

Background:

R. Haggarty, Fundamentals of Mathematical Analysis, Adison Wesley.

M. Spivak, Calculus, Benjamin-Cummings.

D. Stirling, Mathematical Analysis: a fundamental and straightforward approach, Ellis Horwood.

Details of Assessment

The final assessment of this module will consist of 20% coursework and 80% from a one and a half hour examination during the January exam period. The 20% coursework contribution will be determined by students' solutions to four sets of work, one of which will be done in groups. The examination paper will contain 4 questions with full marks on the paper obtainable from 4 complete answers.


next up previous
Next: MC249 Rings and modules Up: Year 2 Previous: MC244 Geometry
S. J. Ambler
11/20/1999