# KL4001 - Real Analysis

## What will I learn on this module?

The module is designed to i) introduce you to the notion of convergence as this applies to sequences, series and functions of one variable; ii) to provide a firm basis for future modules in which the idea of convergence is used; iii) to help you recognize the necessity and power of rigorous argument.

Outline Syllabus:

1) Introduction to propositional logic and sets.
2) Real numbers: equations, inequalities, modulus, bounded sets, maximum, minimum, supremum and infimum.
3) Sequences: convergence, boundedness, limit theorems; standard sequences and rate of convergence, monotone sequences, Cauchy sequences.
4) Series: standard series (geometric, harmonic series, alternating harmonic series, etc ); absolute and conditional convergence; convergence tests.
5) Power Series.
6) Functions: continuity, the intermediate value theorem, the extreme value theorem.
7) Differentiability: basic differentiability theorems, differentiability and continuity, Rolle’s theorem, Lagrange theorem, Taylor’s theorem.
8) Riemann’s Integrability: properties of integrable functions, modulus and integrals, The fundamental theorem of Calculus.

# How will I learn on this module?

You will learn through a combination of lectures and exercise sessions. Lectures give a formal introduction to theoretical aspects of linear algebra and real analysis. You will attend exercise classes throughout the academic year, during which you will work through problems to develop your knowledge and skills, with the support of the tutor.

Assessment is by two formal examinations at the end of each semester. Each component of the examination is worth 50%. Both examinations will cover will assess your knowledge of the discipline along with your problem-solving abilities.

Exam feedback will be provided individually and also generically to indicate where the cohort has a strong or a weaker answer to examination questions. You will receive both written and oral feedback from the in-class test, as well as formative feedback throughout the course, in particular during the exercise classes.

Independent study is supported by further technology-enhanced resources provided via the e-learning portal.

# How will I be supported academically on this module?

Lectures and exercise classes will be the main point of academic contact, providing you with a formal teaching environment for core learning. In particular, exercise classes will provide you with opportunities for critical enquiry and exchanges. Outside formal scheduled teaching, you will be able to contact the module team (module Tutor, year Tutor, Programme Leader) either via email or the open door policy operated throughout the programme. Further academic support will be provided through technology-enhanced resources via the e-learning portal. You will also have the opportunity to give your feedback formally through periodic staff-student committees and directly to the module Tutor at the end of the semester.

# What will I be expected to read on this module?

All modules at Northumbria include a range of reading materials that students are expected to engage with. The reading list for this module can be found at: http://readinglists.northumbria.ac.uk
(Reading List service online guide for academic staff this containing contact details for the Reading List team – http://library.northumbria.ac.uk/readinglists)

# What will I be expected to achieve?

Knowledge & Understanding:
1. To demonstrate the basic concepts and definitions of linear algebra and real analysis (KU1).
2. To apply basic algebraic techniques and fundamental principles of real analysis (KU1).
Intellectual / Professional skills & abilities:

3. Construct rigorous mathematical arguments to produce relatively complex calculations, understanding their effectiveness and range of applicability (IPSA1).
4. Select and apply appropriate algebraic and analytical methods to solve standard linear algebra and real analysis problems (IPSA2).

Personal Values Attributes (Global / Cultural awareness, Ethics, Curiosity) (PVA):
5. Demonstrate the ability of learning new concepts, describing and solving mathematical problems making use of appropriate materials/resources.

# How will I be assessed?

SUMMATIVE
1. Examination component 1 (50%) – 1, 2, 3, 4, 5
Examination component 2 (50%) – 1,2, 3, 4, 5

FORMATIVE

1. Exercise classes – 1, 2, 3, 4, 5

Feedback will take several forms, including verbal feedback during exercise classes; individual verbal and written comments on the in-class test assessment delivered and via blackboard; written feedback on the exam.

N/A

N/A

# Module abstract

Real Analysis will provide you with core mathematical knowledge providing rigorous underpinning to the notions of differentiability and integrability used in Calculus and Vector Calculus. You will be introduced to the fundamental notion of convergence in the contexts of sequences, series and functions of one variable. You will get familiar with a variety of mathematical proof strategies which will provide you with specific skills needed for the rigorous formalisation of mathematical models and analysis of their solutions. The module consists of combination of lectures and exercise classes. During lectures you will be introduced to the definitions and fundamental propositions and theorems that justify basic mathematical operations as well as more advanced concepts of theory of functions and differential calculus. During exercise classes you will be able to formulate and prove mathematical statements experiencing the abstract power of Mathematics and its intrinsic beauty. You will be assessed by two formal examinations at the end of each semester designed to put forward your new skills and techniques. You will receive constructive feedback during exercise classes throughout the year. The eLearning Portal will serve as a point of contact, information and discussion with the tutor.
Concepts and skills you will learn in this module will constitute a solid foundation for your further studies, hone your abstract thinking, sharpen your analytical mindset and nurture your effective problem-solving skills enhancing your employability on the longer term.

# What will I learn on this module?

The module is designed to i) introduce you to the notion of convergence as this applies to sequences, series and functions of one variable; ii) to provide a firm basis for future modules in which the idea of convergence is used; iii) to help you recognize the necessity and power of rigorous argument.

Outline Syllabus:

1) Introduction to propositional logic and sets.
2) Real numbers: equations, inequalities, modulus, bounded sets, maximum, minimum, supremum and infimum.
3) Sequences: convergence, boundedness, limit theorems; standard sequences and rate of convergence, monotone sequences, Cauchy sequences.
4) Series: standard series (geometric, harmonic series, alternating harmonic series, etc ); absolute and conditional convergence; convergence tests.
5) Power Series.
6) Functions: continuity, the intermediate value theorem, the extreme value theorem.
7) Differentiability: basic differentiability theorems, differentiability and continuity, Rolle’s theorem, Lagrange theorem, Taylor’s theorem.
8) Riemann’s Integrability: properties of integrable functions, modulus and integrals, The fundamental theorem of Calculus.

### Course info

UCAS Code G100

Credits 20

Level of Study Undergraduate

Mode of Study 3 years full-time or 4 years with a placement (sandwich)/study abroad

Department Mathematics, Physics and Electrical Engineering

Location City Campus, Northumbria University

City Newcastle

Start September 2020

## Mathematics BSc (Hons)

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