M2: Analysis I - Sequences and Series (2024-25)
Section outline
-
-
The main set of lecture notes are the ones by Vicky Neale, below.
These notes are an expanded version of the notes by Hilary Priestley, also below, which come with several supplementary notes on specific topics.
There is a short document regarding a small handful of updates to the main lecture notes, that I have placed in a separate document:
- Updates to the Main Lecture NotesThere are also notes on two non-examinable topics that I have added:
- Dedekind cuts
- An introduction to open and closed setsAs to exercise sheets, there are:
- 7 exercise sheets, which also come with a Moodle version
- 7 Bonus exercise sheets, which are OPTIONAL, kindly provided by Paul Balister, which cover some extension topics and also contain some more challenging questions. It is up to you to decide whether you have time and intersest to do these, but you will certainly find plenty of fun problems. Not all college tutors will have time or a desire to look at bonus exercises, as there is already a lot to cover in the regular sheets. -
This contains comments and updates to the Main Lecture Notes by Vicky Neale. I have been asked to add these things by your Hilary Term Analysis II lecturer. It involves a few minor changes to some definitions in the main set of notes.
It also contains some non-examinable material which is helpful for the Bonus exercise sheets, in case you wish to attempt those, which I was kindly provided with by your Hilary Term Analysis II lecturer.
-
This contains the cultural remarks made about how one can construct real numbers. Although "Real Numbers" are in the syllabus, the definition and usage of Dedekind cuts is non-examinable, so do not worry if you find this topic more difficult than other topics. The lecture notes I inherited only list the properties of real numbers, but not their definition (in the sense of: an explanation of their existence), so you will not find Dedekind cuts in the main lecture notes.
-
This contains an introduction to the notion of open sets and closed sets, which you will need for Analysis II in Hilary Term.
It is not in the syllabus of Analysis I, that is why it does not appear in the main lecture notes. I have been asked by the Hilary Term Analysis II lecturer to tell you these things at the end of the course, because they will be needed in that course.
-
These were kindly provided by Paul Balister, your Hilary Term Analysis II lecturer.
Although these sheets are optional, you are highly encouraged to at least attempt the following exercises:
Bonus Sheet - 2 - exercises 1 and 2
Bonus Sheet - 4 - exercise 1 (an improved version of the regular sheet 1 ex.4)
Bonus Sheet - 7 - exercises 1 and 2.These sheets cover some extension topics and also contain some more challenging questions. It is up to you to decide whether you have time and intersest to do these, but you will certainly find plenty of fun problems. Not all college tutors will have time or a desire to look at bonus exercises, as there is already a lot to cover in the regular sheets.
-
-
-
The main set of lecture notes are the ones by Vicky Neale, below.
These notes are an expanded version of the notes by Hilary Priestley, also below, which come with several supplementary notes on specific topics.
Under the "General" header you will find the following additional material:
There is a short document regarding a small handful of updates to the main lecture notes, that I have placed in a separate document:
- Updates to the Main Lecture NotesThere are also notes on two non-examinable topics that I have added:
- Dedekind cuts
- An introduction to open and closed setsBeyond the 7 regular exercise sheets (with Moodle version), there are also:
- 7 Bonus exercise sheets, which are OPTIONAL, kindly provided by Paul Balister, which cover some extension topics and also contain some more challenging questions. It is up to you to decide whether you have time and intersest to do these, but you will certainly find plenty of fun problems. Not all college tutors will have time or a desire to look at bonus exercises, as there is already a lot to cover in the regular sheets. -
-
This block of work corresponds to Weeks 1 and 2 (3 lectures), and is designed for tutorials in Week 3. It's about the real numbers, arithmetic, ordering, and modulus. Lecture notes Sections 2, 3, 4, 5, 6, 7.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 1 Moodle version for those who find that more convenient/accessible.
There are also optional activities for this block of work: a quiz (Sheet 1) Starter 1 - Axioms for R and (Sheet 1) Starter 2 - Name that axiom and (Sheet 1) Pudding - Field or Not a Field? . There's also an extra optional quiz Highlights of the Greek alphabet. It's not directly related to this block of work as we shan't use Greek letters till slightly later in term, but it's available to you at any time and might be useful if you don't yet feel confident with naming and writing Greek letters.
-
This block of work corresponds to Week 3, and is designed for tutorials in Week 4. It's about the modulus, supremum, infimum, and countability. Lecture notes Sections 8, 10, 11, 12, 13, 14 (note that there is no section 9).
The problems sheet is below as a pdf attachment, and is also available as a Sheet 2 Moodle version for those who find that more convenient/accessible.
There are also three optional activities for this block of work: (Sheet 2) Starter - Upper and lower bounds, (Sheet 2) Starter - sup and max, and (Sheet 2) Pudding - Countability.
-
This block of work corresponds to Week 4, and is designed for tutorials in Week 5. It's about convergence, infinite limits, and complex sequences. Lecture notes Sections 15, 16, 17, 18, 19, 20.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 3 Moodle version for those who find that more convenient/accessible.
Here are a couple of resources you might find useful. One is self explanation training - if you head to that link you'll find a pdf booklet for students, which might take you about half an hour to work through. The other is How to think about Analysis, by Lara Alcock. This is on the reading list for this course, where you can click through to an electronic version available via the university library.
There is also an optional activity for this block of work: (Sheet 3) Pudding - Mandelbrot set.
-
This block of work corresponds to Week 5, and is designed for tutorials in Week 6. It's about subsequences, the algebra of limits, and monotonic sequences. Lecture notes Sections 21, 22, 23, 24, 25.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 4 Moodle version for those who find that more convenient/accessible.
There are also three optional activities for this block of work: (Sheet 4) Starter - Sequences and subsequences, (Sheet 4) Starter - Big O and little o notation, and (Sheet 4) Pudding - Limit of limits.
-
This block of work corresponds to Week 6, and is designed for tutorials in Week 7. It's about Cauchy sequences, series (convergence and absolute convergence), properties of e, and the Alternating Series Test. Lecture notes Sections 26, 27, 28, 29, 30.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 5 Moodle version for those who find that more convenient/accessible.
There are also two optional activities for this block of work: (Sheet 5) Starter - Bolzano-Weierstrass Theorem and (Sheet 5) Pudding - Some sequences and series .
-
This block of work corresponds to Week 7, and is designed for tutorials in Week 8. It's about convergence of series and tests for convergence. Lecture notes Sections 31, 32, 33, 34.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 6 Moodle version for those who find that more convenient/accessible.
-
This block of work corresponds to Week 8, and is designed for tutorials at the start of Hilary Term. It's about conditionally convergent series and power series. Lecture notes Sections 34, 35, 36, 37.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 7 Moodle version for those who find that more convenient/accessible.
There is also one optional activity for this block of work: (Sheet 7) Pudding - Stirling's formula.
-
-
-
This quiz is designed to help you learn the letters from the Greek alphabet that you are most likely to need as a mathematician. You can tackle it at any stage of the term.
-