General
Main content blocks
- Lecturer: Profile: Alexander Ritter
\(\bullet\) prove results within an axiomatic framework;
\(\bullet\) define and prove basic results about countable and uncountable sets, including key examples;
\(\bullet\) define what it means for a sequence or series to converge;
\(\bullet\) prove results using the completeness axiom for \(\mathbb{R}\) and using Cauchy’s criterion for the - convergence of real and complex sequences and series, and explain how completeness and Cauchy’s criterion are related;
\(\bullet\) analyse the convergence (or otherwise) of a variety of well known sequences and series, and use this to conjecture the behaviour of unfamiliar sequences and series;
\(\bullet\) apply standard techniques to determine whether a sequence converges, and whether a series converges;
\(\bullet\) define the elementary functions using power series, and use these definitions to deduce basic properties of these functions.
Sequences of real or complex numbers. Definition of a limit of a sequence of numbers. Limits and inequalities. The algebra of limits. Order notation: \(O\), \(o\).
Subsequences; a proof that every subsequence of a convergent sequence converges to the same limit; bounded monotone sequences converge. Bolzano--Weierstrass Theorem. Cauchy's convergence criterion.
Series of real or complex numbers. Convergence of series. Simple examples to include geometric progressions and some power series. Absolute convergence, Comparison Test, Ratio Test, Integral Test. Alternating Series Test.
Power series, radius of convergence. Examples to include definition of and relationships between exponential, trigonometric functions and hyperbolic functions.
Course Syllabus:
See the examinable syllabus (https://canvas.ox.ac.uk/courses/64592/pages/synopses-and-syllabus-prelims).
Section outline
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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.
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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.
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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.
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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.
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