MTH5104 - Convergence and Continuity - 2023/24
Topic outline
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MTH5104 - Convergence and Continuity
Module Organiser: Dr Navid Nabijou
Year: 2023-2024 | Semester: A | Level: 5 | Credits: 15Classes:
Lectures: Mondays 16:00-18:00 (Fogg Lecture Theatre)
Tutorials: Fridays 14:00-16:00 (Fogg Lecture Theatre)
Asynchronous classes: Q-Review recordings posted weekly.
Q-Review will allow live broadcast, but this will be a decidedly inferior experience. You are strongly implored to attend the lectures in-person.Learning Support:
Assessment:
Learning Support Hour: Mondays 14:00-15:00 at the Learning Café in the SMS Social Hub (MB-B11). Every week except Week 7.
80% final exam
20% in-term coursework (constituting 5 QMPlus tests throughout the semester)
For full assessment information, click on the "Assessment" tab at the top of this page. -
Module Organiser: Dr Navid Nabijou
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For assessment information, click the "Assessment" tab at the top of the page.
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This is a module which rewards active engagement. Each week, you are expected to:
- Participate in the lectures on Monday and Friday (4 hours).
- Study the relevant parts of the lecture notes (2 hours).
- Attempt the problem sheets (3 hours).
In addition there will be 5 courseworks, each counting for 4% of your grade, staggered throughout the term. These will be administered as QMPlus quizzes. See the "Assessment" tab at the top of the module page.
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It is likely that at certain points you will get stuck working on the problem sheets, or are puzzled by something covered in the module. This is completely normal, and the discomfort you feel when this happens is actually an important part of the learning process. You don't have to face these difficulties alone. You can:
- Post on the module's Student Forum.
- Ask the lecturer for help in the next lecture or tutorial.
- Attend the next Learning Support Hour.
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Module description
Convergence and continuity are two of the basic concepts of Real Analysis, the area of mathematics that provides the theoretical foundations of Calculus.
In this module we start by exploring the algebraic and order properties of the real numbers, properties that are the bread and butter of Calculus as well as Real Analysis.
We then build on our working knowledge of convergence and continuity, starting by formalizing the familiar expression, “…getting closer and closer to…”. We shall see that the rigorous mathematical formulation to this idea, is based on the concept of distance. The latter plays an important role in the definitions of convergence and continuity.
Equipped with the concepts of convergent sequences, and continuous functions we shall answer questions such as: What does it mean to say that a series converges to a limit? Are there kinds of function that are guaranteed to have a maximum value?
Real Analysis is a beautiful and important branch of pure mathematics, and this module is a first introduction, with many examples, to it.
The material covered in this module is essential for further pure mathematics modules such as “Differential and Integral Analysis” and “Metric Spaces and Topology”.
SYLLABUS
- Real numbers: Algebraic properties, inequalities, supremum and infimum, completeness axiom for the existence of the supremum.
- Sequences: Definition of limit and its use in specific examples, limit of sum, product and quotient of sequences. Bounded monotone sequences. Statement of the Bolzano-Weierstrass Theorem.
- Series: Convergent series, geometric series, harmonic series. Comparison and ratio tests. Absolutely convergent series. Power series. Examples, including sin(x), cos(x) and exp(x).
- Continuous functions: Definition of continuity and its use in specific examples, sum of continuous functions, composition of continuous functions (proofs), products/quotients of continuous functions (stated). The Intermediate Value Theorem, application to roots of polynomials, boundedness of continuous functions on closed bounded intervals.
Learning Aims and OutcomesDISCIPLINARY SKILLS
At the end of this module, students should be able to:
- Define the basic concepts underlying continuous mathematics: supremum, limit of a sequence, convergent series, continuous function, derivative.
- Use criteria for convergence of series and continuity of functions.
- Prove general results concerning infinite sequences, convergent series and continuous functions.
- Solve problems relating to convergence of series and continuity of functions.
ATTRIBUTES
At the end of this module, students should have developed with respect to the following attributes:- Grasp the principles and practices of their field of study.
- Explain and argue clearly and concisely.
- Apply their analytical skills to investigate unfamiliar problems.
- Acquire substantial bodies of new knowledge.
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Revision
To prepare for the exam, you should:- Make sure you are comfortable with the key definitions, important theorems, and main examples from the lecture notes.
- Attempt all the questions on the problem sheets. Some of these are a lot harder than would be found on an exam, but if you stretch yourself in this way, the exam itself will be easy.
- Attempt the practice exam (see below), and past exams. The practice exam in particular should give you a good idea of the level expected for the exam.
Exam Rules
Here are some useful "ground rules" for the exam:- The typed lecture notes can be considered a complete reference. No exam question will require you to know definitions or theorems not contained in those notes.
- Your mathematical arguments should be written in full sentences. Try to emulate the style of proofs given in the lecture notes, in the tutorials, in the solutions to the problem sheets, and in the practice exam.
- Unless a question explicitly asks you to work "directly from the definition", you can use any result from lectures, provided that you state it clearly. This last proviso is important. A similar message will be contained on the exam itself.
You can find the practice exam and solutions at the bottom of this block. This is the best preparation for the exam, so make sure you use this resource strategically: only attempt it once you are comfortable with the material and have worked hard on the problem sheets. Model solutions are provided to the exam; I am not able to mark individual exams, but there will be a couple of Learning Cafes in early January (dates TBA) where you can ask me questions.Past Exams
The past exam repository is a useful resource. Solutions to some, but not all, of the past exams are posted at the bottom of this block. The module has changed a little bit over the years. You will have to use your own judgement as to which questions in the past exams are still relevant. -
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Forum Description: This forum is available for everyone to post messages to. Students can raise questions or discuss issues related to the module. Students are encouraged to post to this forum and it will be checked daily by the module leaders. Students should feel free to reply to other students if they are able to.
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At the top of this page you will see two rows of tabs, one above the other. Here are the most important ones:
- Module Content (second row, first entry): lecture notes, problem sheets, past exams will be uploaded here.
- Q-Review (second row, last entry): here you can access recordings of the lectures and tutorials.
- Assessment (first row, second entry): here you can find detailed information on the assessment for the course. It is also where the in-course quizzes will appear.
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Full lecture notes are available here. The notes will be tweaked as the module progresses. (This may mean that theorem numbers change; this is the price you pay for having access to all the notes from the start.)
These can be considered a complete reference for the module; only material contained in the lecture notes is examinable.
For reference, you can also find the lecture notes from last year. For the avoidance of doubt, only material in the current (2023-2024) notes is examinable.
Below is a schedule, which roughly outlines the correspondence between each week's lectures and the lecture notes. It will be updated as the semester progresses:
Week Material 1 Introduction, 1A-1E 2 2A-2C 3 2D 4 3A-3B 5 3C-3D 6 3D-4A 8 4B-4D 9 4E-5A 10 5A-5B 11 5C-5D 12 5E -
Lecture notes for the current (2023-2024) iteration of the course. Please inform me of any typos.
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Overview of the current (2023-2024) iteration of the course. This list is for illustration purposes only and is non-exhaustive: for a full list of topics, consult the lecture notes above.
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Lecture notes from the previous (2022-2023) iteration of the course. The material covered is similar, but not identical, to the current course.
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The problem sheets are a central part of the course, not an optional extra! It is not possible to understand the material without grappling with problems (this is equally true for professional mathematicians like myself).
Problem sheets will appear here in due course. Solutions will appear approximately two weeks after. In each tutorial, we will attempt some problems from the problem sheets.
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For assessment information, click the "Assessment" tab at the top of the page.
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Assessment pattern. 80% final assessment, 20% in term assessment.
Final assessment format (80%). Your final assessment will be an on campus exam, held during the January examination period. It will be 3 hours in duration, with SpLD accommodations handled separately. Calculators and outside notes will not be permitted.
In-term assessment format and dates (20%). Your in-term assessment will consist of 5 online quizzes (each contributing 4% towards your total grade), administered online via QMPlus. They will appear in the "Assessment" tab at the top of the page. Each quiz will be available on QMPlus for a 24 hour period. When you begin the quiz a timer will start running, and from that moment you will have 1 hour to complete the quiz. Late submissions are not permitted, and will receive zero marks. (Quiz 4 is an exception. This will be open for 24 hours, with no 1 hour countdown.)
Important. Each quiz is available for a generous 24 hour period. Do not leave your attempt until the last hour. If you do this, we will not be able to help you if you run into any technical issues. You have been warned!
The dates and times of the quizzes are as follows. Each quiz takes place on a Wednesday, and will involve material up to and including the preceding Friday:
Week Opens Closes Topics Format Special Notes 4 14:00 on Wednesday 18th October 14:00 on Thursday 19th October Real Numbers (Chapter 2) Multiple choice 6 14:00 on Wednesday 1st November 14:00 on Thursday 2nd November Sequences (Chapter 3) Handwritten 8 14:00 on Wednesday 15th November 14:00 on Thursday 16th November Sequences (Chapter 3) Multiple choice 10 14:00 on Wednesday 29th November 14:00 on Thursday 30th November Series (Chapter 4) Handwritten Open for full 24 hours; no 1 hour countdown 12 14:00 on Wednesday 13th December 14:00 on Thursday 14th December Continuity (Chapter 5) Multiple choice Feedback. Feedback on the problem sheets is available every week, during the Tutorial and the Learning Support Hour.
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There are many good books available. Search for introductory texts in real analysis. An absolute classic is:
- Walter Rudin. Principles of Mathematical Analysis. (Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976.)
If you are interested in expanding your horizons, a very interesting text is:
- Bernard Gelbaum and John M. H. Olmsted. John M. H. Counterexamples in analysis. (Corrected reprint of the second (1965) edition. Dover Publications, Inc., Mineola, NY, 2003.)
Not strictly mathematics, but the following graphic novel features a lively cast of characters and centres on the crisis in the foundations of mathematics in the early 20th century. It does a good job of conveying the heroism inherent to the life of a mathematician.
- Apostolos Doxiadis and Christos H. Papadimitriou. Logicomix: An Epic Search for Truth. (Bloomsbury Publishing PLC; UK ed. edition, 2009.)
