MST20040 Analysis

Academic Year 2020/2021

Analysis is a module designed to introduce to students some of the theory developed on sequences and series in the 19th century.
The student will be introduced to the concept of a sequence of real numbers and will learn about various other concepts such as
that of convergent sequence, bounded sequence and monotonic sequence to name a few. The Axiom of Completeness will be introduced and the student will see its role in proving results such as the Monotone Convergence Theorem and Bolzano-Weierstrass Theorem. The concepts of countable and uncountable sets will also be discussed and the student will learn that the set of real numbers is uncountable. The latter part of the module deals with tests of convergence for series and looks at the concepts of conditional and absolute convergence. The module concludes with a discussion of power series and Taylor series.

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Curricular information is subject to change

Learning Outcomes:

On completion of this module the student should be able to:

Identify, define, graph (if relevant), and generate examples of the major concepts of sequences and series of real numbers.

Describe and give examples of the relationships between the major concepts of sequences of real numbers.

Verify the veracity of statements about sequences of real numbers and support your answer with a mathematical argument.

State, apply, prove and describe the main theorems relating to sequences of real numbers.

Describe and apply the Axiom of Completeness.

Define and describe the concepts of countable and uncountable sets.

Test a given series for convergence.

Find the interval of convergence of a power series.

Find the Taylor series generated by a given function.

Indicative Module Content:

The following is indicative curricular content. Minor changes may be made throughout the semester.

Section 1. Introduction and Preliminaries

1.1. What is this course about?
1.2 Preliminaries.

Section 2. Sequences - a First Look

2.1. Sequences.
2.2. Null Sequences.
2.3. Convergent Sequences.

Section 3. The Real Numbers - Completeness

3.1. The Axiom of Completeness.
3.2. Consequences of Completeness.

Section 4. Sequences - a More Indepth Look

4.1. The Monotone Convergence Theorem.
4.2. The Bolzano-Weierstrass Theorem.
4.3. Cauchy Sequences and Convergence.

Section 5. Infinite Series

5.1. Infinite Series -- An Introduction.
5.2. Geometric Series.
5.3. Series with Nonnegative Terms.
5.4. Series with Positive and Negative Terms.

Section 6. Power Series

6.1. Introduction.
6.2. Power Series.
6.3. Taylor Series.

Student Effort Hours: 
Student Effort Type Hours
Lectures

36
Tutorial

10
Specified Learning Activities

22
Autonomous Student Learning

57
Total

125
Approaches to Teaching and Learning:
All lectures will be online with some live and some pre-recorded, depending on the nature of the material. Generally the live lectures will be interactive, with students working in small groups. Where this is the case they will not be recorded to ensure students feel happy about participating freely. Workshops will also be online. There will be a strong emphasis on active-learning in workshops with online videos to help student prepare for them. Again they will not be recorded to ensure students feel happy to participate. In this module you will be expected to engage throughout as a lot of the learning takes place using the weekly activities. You will also be expected to study independently and to prepare for workshops. 
Requirements, Exclusions and Recommendations
Learning Requirements:

Students must have a strong foundation in Calculus. For examples, students who have completed modules such as MST10010 and MST10020 OR MATH10350 OR MATH10130 and MATH10140 have the prerequisites for this module.


Module Requisites and Incompatibles
Incompatibles:
MATH10320 - Mathematical Analysis, MATH20170 - Introduction to Analysis


 
Assessment Strategy  
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Portfolio: Portfolio of Examples which will be completed within the first six weeks. Varies over the Trimester n/a Standard conversion grade scale 40% No

20
Continuous Assessment: An end-of-trimester reflection on what you learned. Coursework (End of Trimester) n/a Standard conversion grade scale 40% No

10
Class Test: Decide if given statements are true or false and verify your answer. Held during a lecture time and will be open book. Week 9 n/a Standard conversion grade scale 40% No

30
Continuous Assessment: A maximum of 10% will be given for working on designated "Activities" in throughout the semester. There is 1% per Activity and twelve will be given in total. Throughout the Trimester n/a Standard conversion grade scale 40% No

10
Examination: Written examination. Content from the last four weeks of the module will be examined. 1 hour End of Trimester Exam Yes Standard conversion grade scale 40% No

30

Carry forward of passed components
No
 
Resit In Terminal Exam
Autumn Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 
Feedback Strategy/Strategies

• Feedback individually to students, on an activity or draft prior to summative assessment
• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment
• Peer review activities

How will my Feedback be Delivered?

Group-feedback will be provided on all Activities. Summative feedback will be provided individually on the Class Test and formative group feedback will be provided. Summative feedback will be provided individually on the Portfolio of Examples, with formative group feedback and a peer-assessment activity will also be conducted. Summative feedback will be provided on the final examination and end-of-trimester reflection.