Section: Course Materials | M2: Analysis I - Sequences and Series (2024-25)

Course Term: Michaelmas

Course Lecture Information: 15 lectures

Course Overview:

In these lectures we study the real and complex numbers, and study their properties, particularly completeness (roughly speaking, the idea that there are no 'gaps' - unlike in the rational numbers, for example). We go on to define and study limits of sequences, convergence of series, and power series.

Learning Outcomes:

By the end of the course, students will be able to:

\(\bullet\) 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.

Course Synopsis:

Real numbers: arithmetic, ordering, suprema, infima; the real numbers as a complete ordered field. Definition of a countable set. The countability of the rational numbers. The reals are uncountable. The complex number system. The triangle inequality.

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