University of Manchester

Mike Prest

Teaching

**Office Hour: **If you want to discuss something with me then send me an email and we can arrange a time. You can also ask questions by email (as well as in person). If you turn up at my office and I am free and not in the middle of something that I need to push on with, then I will be happy to talk to you/answer your questions. We are supposed to set an office hour; mine is Monday 12.30-1.30.

My room is Alan Turing Building 1.120

Internal tel. 55875

email mprest@manchester.ac.uk

MATH10242 Sequences and Series

**Notes for the course:** These online notes are fairly complete and I will cover the material pretty much in this order. But, in lectures, I will present things a little differently, concentrating on the main ideas and I will give additional examples, as well as more explanation. So, if you miss a lecture, try to get hold of the notes from someone who was there (also: I will use mostly the visualiser rather than the board, so most of what I write (and say) will be recorded). (Online notes updated 15:10 on 26th January: minor numbering corrections, as follows: page 19, line -15 now Question 5 [not 3] (of the Problems sheet for Week 2); page 20, line 3 now Question 4 from the Problems sheet for Week 3; pages 73 (1 occurrence) and 74 (3 occurrences) Question 3 [not 2] on the Week 11 Exercise sheet. Any further changes/corrections I will write in red on the pdf.)

I did say that I would give a proof of the fact that the limit of (1+1/n)^n as n goes to infinity is e; here is a short, elementary proof taken from the web.

**Getting feedback on your work:** Doing examples: both the weekly example sheets and the lecture notes contain many exercises; you should write out your own solutions to these and *then* compare with the solution sheets, respectively what's written in the lecture notes. That comparison is one form of feedback and the coursework test+solutions is similar. The most detailed feedback on your understanding and on how you write mathematics is obtained by asking questions in the tutorials and/or in my office hour.

Response to Week 3 Feedback comments

Response to UEQ (Feedback) comments

Exercise sheets, solutions and other material will be added here as the course progresses.

Visualiser slides from Week 1 Lectures

Visualiser slides from Week 2 Lectures

Visualiser slides from Week 3 Lectures

Visualiser slides from Week 4 Lecture

Guide to reading/exercises in Weeks 4 and 5. Read this to see what you should be aiming to do for this course unit in Weeks 4 and 5.

Visualiser slides from Week 5 Lecture

Guide to reading/exercises in Week 6.

Guide to reading/exercises in Week 7. Read this to see what you should be aiming to do for this course unit in Week 7.

Visualiser slides from Week 8 Lecture

Visualiser slides from Week 9 Lectures The first 4 pages are last year's visualiser notes (since, on the Tuesday it was not possible to display the visualiser input).

Visualiser slides from Week 10, first Lecture

Visualiser slides from Week 11 Lectures

Visualiser slides from Week 12 Lectures (2nd lecture; no specific notes from the first, summary/review, lecture)

**Some Visualiser Slides from last year's lectures** These might be helpful in working on the materials in Chapters 3 through 7 of the course notes.

Visualiser slides from 2017 Week 3, 2nd Lecture

Visualiser slides from 2017 Week 4 Lectures

Visualiser slides from 2017 Week 5 Lectures

Visualiser slides from 2017 Week 6 Lectures

Visualiser slides from 2017 Week 7 Lecture

**Weekly Exercise and Solution Sheets**

Exercises for Week 2 tutorials

Solutions to exercises for Week 2 tutorials

Exercises for Week 3 tutorials

Solutions to exercises for Week 3 tutorials (Q3(a) beginning of 3rd line of solution should read 1/epsilon, not 1/e)

Exercises for Week 4 tutorials

Solutions to exercises for Week 4 tutorials

Exercises for Week 5 tutorials

Solutions to exercises for Week 5 tutorials

Exercises for Week 6 tutorials

Solutions to exercises for Week 6 tutorials

Exercises for Week 7 tutorials

Solutions to exercises for Week 7 tutorials

Week 8 Review questions with answers (answers should show as blue writing; use a different browser or download the file if not visible)

Exercises for Week 8 tutorials

Solutions to exercises for Week 8 tutorials

Exercises for Week 10 (the file says Week 9) tutorials

Solutions to exercises for Week 10 (the file says Week 9) tutorials

Exercises for Week 11 tutorials

Solutions to exercises for Week 11 tutorials

Exercises for Week 12 tutorials

Solutions to exercises for Week 11 tutorials

Extra Exercises This is a large set of exam-style questions.

**Coursework Test** Held on Tuesday 20th March.

2018 Coursework Test and commented Solutions to 2018 Coursework Test The test was 35 minutes duration.

**Practice Coursework Tests**

2014 Coursework and Solutions to 2014 Coursework

2016 Coursework and Solutions to 2016 Coursework

2017 Coursework Test and commented Solutions to 2017 Coursework The test was 35 minutes duration.

**Some past Exam Papers** If you spot any typos or ambiguities in the solutions, let me know.

2014 Exam and Solutions to 2014 Exam

2015 Exam and Solutions to 2015 Exam

2016 Exam and Solutions to 2016 Exam

Feedback comments on 2017 Exam

Feedback comments on 2018 Exam

**About this year's exam** The format will be the same as previous years and the structure/type of questions will be rather similar. Students sometime ask what proofs they need to know - you can take previous years' exams as a good guide to that, from which you can see that there is rather little point in memorising long proofs. You *do* need to know the basic definitions and results - again, previous years' exams and solutions are a good guide.

**Review session, 2-4 on Monday 4th June, in Kilburn Theatre 1.1:** I will go through examples from past papers mainly, I expect, in response to specific requests from you. Here are the visualiser scans from the review session.

Some **sources for further reading**:

HELM Mathematics Workbooks - These can be useful for illustrating and reinforcing some of the basic ideas; Workbook 16 is Sequences and Series (mostly series; it says a little about sequences but doesn't cover very much).

There's a lot in the JRUL (John Rylands University Library) and you can find sets of notes on the web, such as:

E. Zakon, Basic Concepts in Mathematics, available as a pdf at http://www.trillia.com/zakon1.html

There are a number of other good free web-books available, many of which can be found on the e-books website:

http://www.e-booksdirectory.com/listing.php?category=471

All of the following are available as hard copy from the JRUL; a couple are also available in e-book format.

Analyis Books:

R. Haggarty, Fundamentals of Mathematical Analysis, Prentice Hall, 1993. ISBN-10:0201631970

V. Bryant, Yet Another Introduction to Analysis, Cambridge University Press, 1990. ISBN-10:052138835X (also available as an ebook through JRUL)

L. Alcock, How to Think about Analysis, Oxford University Press, 2014. ISBN-13:978-0198723530 (also available as an ebook through JRUL)

Books covering more topics:

P.J. Eccles, An Introduction to Mathematical Reasoning: Numbers, Sets and Functions, Cambridge University Press, 1997. ISBN-10:0521597188

G.C. Smith, Introductory Mathematics: Algebra and Analysis, Springer, 2008. ISBN-10:3540761780

MATH43051/63051 Model Theory This course has been modified a bit since last year, the main change being the inclusion of ultraproducts and Los' Theorem.

**Outline**

Weeks 1,2,3: Ultraproducts and Los' Theorem

Week 4,5: Theories and their models; Downwards Lowenheim-Skolem

Weeks 7,8: Dense Linear Orders; Back-and-forth arguments; The Random Graph

Weeks 9,10,11: Types; realising types; saturated models; the space of types; Upwards Lowenheim-Skolem

Week 12: Aleph_0 categoricity

**Notes for the course:** here are the notes for Sections 1-3, for Sections 4-7 (you can ignore Sections 4.4 and 5.1; now, 22nd November, with some handwritten comments, see 4.15, 6.4, 6.13, 6.17) and for the Appendix on Languages and Structures. These notes contain definitions, statements of results, proofs of those (in some cases in full detail, in other cases just the outline) plus various comments and illustrations/exercises. In the lectures I will give some more details and further examples, as well as more explanation. The material in the Appendix is kind of assumed background from Predicate Logic; `kind of' because not all of you will have done a course in Predicate Logic, so I will introduce the material that I need, but briefly and a bit informally. Sometimes the informal idea will be enough but sometimes you will have to refer to the Appendix for the precise definitions and the details.
If you spot any typos, errors, ... in the notes or elsewhere then do let me know.

**Exercises**

**Review exercises**: You can ignore these if you're happy with the ideas from Predicate Logic; they provide some practice if you're not. The first three sets are short exercises on the basics of predicate languages; the set after that contains rather longer exercises.

Exercises on Terms, Formulas and Sentences (with answers on the second page)

Exercises on Reading formulas in Structures (with answers on the second page)

Exercises on Relations and Definable Sets (with answers on the second page)

Exercises on Languages and Interpretations and with answers and the construction tree.

**Exercises**

First Set of Model Theory Exercises

Solutions to First Set of Model Theory Exercises

Second Set of Model Theory Exercises

Solutions to Second Set of Model Theory Exercises

Third Set of Model Theory Exercises

Solutions to Third Set of Model Theory Exercises

Fourth Set of Model Theory Exercises

Solutions to Fourth Set of Model Theory Exercises

**Courseworks**

Coursework 1 Hand in by 4 p.m. on Thursday 26th October.

Coursework 2 Hand in by 4 p.m. on Thursday 30th November.

(Optional) Coursework 3 To be received by the end of Monday 8th January. If you do all three courseworks, then the best two marks will count.

**Courseworks from 2016**

(Optional) Coursework 3 If you do all three courseworks, then the best two marks will count.

**Courseworks from 2015**

(Optional) 2015 Coursework 3 If you do all three courseworks, then the best two marks will count.

**Exams**
The examination for the course is 3 hours duration; you should answer 3 out of 4 questions (if you answer all four, your best three answers will be counted).

Here are general comments on how the January 2018 exam went.

Past exam papers should be on the university website . And here are (partial) solutions to the January 2015 exam and some general comments and solutions to the January 2016 exam. And, finally, some general comments on the January 2017 exam.

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