# REAL ANALYSIS 1 - 2021/2

Module code: MAT1032

## Module Overview

This module is an introduction to analysis, which is the branch of mathematics that rigorously studies functions, continuity and limit processes, such as differentiation and integration. The module leads, among other things, to a deeper understanding of what it means for a sequence or series to converge. Tools such as convergence tests are presented and their validity proved, and the rigorous use of definitions and logic play a central role.   This course lays the foundations for the Year 2 module in Real Analysis 2 (MAT2004) in particular, and, more generally, underpins other modules where a culture of rigorous proof exists.

### Module provider

Mathematics

BEVAN Jonathan (Physics)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 95

Seminar Hours: 11

Guided Learning: 11

Captured Content: 33

Semester 1

None.

## Module content

Indicative content includes:

• Irrational, algebraic and transcendental numbers.

• The axioms of real numbers. Denseness of rational and irrational numbers. Maximum, minimum, supremum and infimum of sets, sequences and functions. The triangle inequality.  Simple estimates.

• Natural induction, set notation, cardinalities of sets (in particular the rationals and reals)

• Axiom of Completeness, and its consequences for the existence of limits. Role of quantifiers in stating and verifying mathematical definitions.

• Sequences: convergence, and other properties. Boundedness, Cauchy sequences, subsequences and the Theorem of Bolzano-Weierstrass.

• Infinite series, convergence and absolute convergence. Convergence tests, including proofs.  Power series, radius and region of convergence.

## Assessment pattern

Assessment type Unit of assessment Weighting
Online Scheduled Summative Class Test ONLINE TEST 25
Examination Online ONLINE EXAM 75

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

·         Understanding of and ability to interpret and manipulate mathematical statements.

·         Subject knowledge through the recall of key definitions, theorems and their proofs.

·         Analytical ability through the solution of unseen problems in the test and exam.

Thus, the summative assessment for this module consists of:

·         One two hour examination (two of three best answers contribute to exam mark, with Question 1 compulsory) at the end of Semester 1; worth 75% module mark.

·         One in-semester test; worth 25% module mark.

Formative assessment and feedback

Students receive written feedback via a number of marked coursework assignments over an 11 week period.  In addition, verbal feedback is provided by lecturer/class tutor at biweekly seminars and weekly tutorial lectures.

## Module aims

• Introduce students to quantifiers, logical statements, countability, suprema, maps, sequences and series.
• Enable students to determine limits of sequences, to determine countability, and to test and prove the convergence of series and sequences.
• Illustrate the application of various techniques for solving frequently encountered problems in analysis.

## Learning outcomes

 Attributes Developed 001 Demonstrate understanding of the real numbers, their axioms and the role of completeness in the existence of limits and solutions to equations. K 002 Interpret and apply quantifiers in mathematical statements, and quote and apply basic theorems in analysis. KCT 003 Calculate limits of sequences and (power) series, and prove/disprove convergence using the definitions KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A detailed introduction to the real numbers, sequences, series and basic ideas of convergence

• Experience (through demonstration) of the methods used to interpret, understand and solve problems in analysis

The learning and teaching methods include:

• 3 x 1 hour lectures per week x 11 weeks, with projector-displayed written notes to supplement the module handbook and Q + A opportunities for students.

• 1 x 1 hour interactive problem solving session/tutorial lecture per week x 11 weeks.

• (every second week) 1 x 1 hour seminar for guided discussion of solutions to problem sheets provided to and worked on by students in advance

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.