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 Tuesdays 12.30-1.30 (but not Tuesday 5th March).

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 (white on blue version) 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).

**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

Exercise sheets, solutions and scans of the visualiser sheets 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 Lectures

Visualiser slides from Week 5 Lectures

Week 6: see the podcasts (video and visualiser input)

Visualiser slides from Week 7 Lecture

Visualiser slides from Week 8 Lectures

Visualiser slides from Week 9 Lectures (2/4/19 Condition added in definition of alternating sequence on last page)

Visualiser slides from Week 10 Lectures

**Weekly Exercise and Solution Sheets**

Exercises for Week 2 tutorials (white on blue version)

Solutions to exercises for Week 2 tutorials (white on blue)

Exercises for Week 3 tutorials (white on blue)

Solutions to exercises for Week 3 tutorials (white on blue) (correction 12/3: p.1, last line, "n" on left replaced by "0")

Exercises for Week 4 tutorials (white on blue)

Solutions to exercises for Week 4 tutorials (white on blue)

Exercises for Week 5 tutorials (white on blue)

Solutions to exercises for Week 5 tutorials (white on blue)

Exercises for Week 6 tutorials (white on blue)

Solutions to exercises for Week 6 tutorials (white on blue)

Exercises for Week 7 tutorials (white on blue)

Solutions to exercises for Week 7 tutorials (white on blue)

Exercises for Week 8 tutorials (white on blue)

Solutions to exercises for Week 8 tutorials (white on blue)

Exercises for Week 9 tutorials (white on blue)

Solutions to exercises for Week 9 tutorials (white on blue)

Exercises for Week 10 tutorials (white on blue)

Solutions to exercises for Week 10 tutorials (white on blue)

Exercises for Week 11 tutorials (white on blue)

**Class Test**

The Class Test 2019 (white on blue) was held on Wednesday 13th March at the usual lecture time. The duration was, in fact, 40 minutes. Here are the commented solutions (white on blue)

**Three Practice Class Tests**

2016 Class Test and Solutions to 2016 Class Test.

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

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

**Three Practice Exam Papers**

2014 Exam and Solutions to 2014 Exam

2015 Exam and Solutions to 2015 Exam

2016 Exam and Solutions to 2016 Exam

And the 2017 exam with Feedback Comments and the 2018 exam with Feedback Comments.

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 (as at early 2018) 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-->

MATH42041/62041 Noncommutative Algebra

**Notes for the course:** here are the notes for Sections 0 and 1, Section 2,
Section 3,
Section 4 (slightly modified, 15/11/18),
Section 5 (alternative proof of 5.16 added, 7/12/18) and
Section 6.
Also see the **slightly revised course notes below** - that file also has comments on what is/isn't examinable.

**Exercises:**

Set of Review Exercises and their Solutions

Exercises 1 and their Solutions

Exercises 2 and their Solutions

Exercises 3 and their Solutions

Exercises 4 and their Solutions

Exercises 5 and their Solutions

Exercises 6 and their Solutions

Exercises 7 and their Solutions

Exercises 8 (6/12/18 Q3(ii) one number changed; the solution sheet will have the solutions to both versions) and their Solutions.

Coursework and Coursework Solutions

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

There will be a **Review Session**, on Tuesday 15th January, 2-4, in Frank Adams Room 1 (1st floor of AT Building, near the kitchen).
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**Student feedback**
Thank you for your, generally very positive, comments on the course unit. A number of you did comment that there is a lot of material in the notes and exercise sheets (indeed there is) and that it would be useful, especially since this is quite hard material, if it were clearer what is core, in particular examinable, and what is "additional material/further reading/extension exercises". I will revise the notes etc. for next year so as to make that clearer.

Comments on January 2019 Exam.

January 2019 Exam and (Brief) Solutions to January 2019 Exam.

Revised Course Notes (complete). I have added some summary comments at the start of sections and some notes indicating what could/would not be examined (at the end of the sections, except in Section 6 where this is at the start). This new text is in blue, as are a few further comments/modifications to the notes (I have made some minor rephrasings etc. but these are not marked).

**Some past exams**

January 2016 Exam and Solutions to and comments on January 2016 Exam

January 2015 Exam and Solutions to and comments on January 2015 Exam

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