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:
Face-to-face lectures but with a strong emphasis on active-learning; interactive workshops; online videos to prepare for workshops. In this module you will be expected to attend, and engage in, lectures and workshops as a lot of the learning takes place using activities in these environments. 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
Class Test: Decide if given statements are true or false and verify your answer. Held in a lecture time. Week 9 n/a Standard conversion grade scale 40% No

20

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: A maximum of 10% will be given for working on "Inclass Exercises" in lectures throughout the semester. There is 1% per Inclass Exercise and twelve will be given in total. Throughout the Trimester n/a Standard conversion grade scale 40% No

10

Examination: Closed-book, written examination. All content from the module is examinable for this exam. 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

50


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 in lectures on all Inclass Exercises. Summative feedback will be provided individually on the Class Test and formative group feedback will be provided in lectures. Summative feedback will be provided individually on the Portfolio of Examples and a peer-assessment activity will also be conducted. Summative feedback will be provided on the final examination.

Timetabling information is displayed only for guidance purposes, relates to the current Academic Year only and is subject to change.

 
Spring
       
Lecture Offering 1 Week(s) - 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31 Fri 09:00 - 09:50 Face to Face
Lecture Offering 1 Week(s) - 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31 Mon 10:00 - 10:50 Face to Face
Lecture Offering 1 Week(s) - 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31 Wed 10:00 - 10:50 Face to Face
Tutorial Offering 2 Week(s) - 20, 21, 22, 23, 24, 27, 28, 29, 30, 31 Thurs 11:00 - 11:50 Face to Face
Tutorial Offering 3 Week(s) - 20, 21, 22, 23, 24, 27, 28, 29, 30, 31 Tues 11:00 - 11:50 Face to Face
Spring