Informal course description

This doesn't replace the synopsis (which formally describes the examinable content of the course).  But often the synopsis is a bit cryptic until after you've studied the material, and I hope that an informal course description might help you to see what you can look forward to later in the term.

The idea of a limit is fundamental and pervasive in maths.  Precise definitions of limits, in different circumstances, underpin the area traditionally known as (mathematical) analysis.  The first main aim of this course is to define what it means for a sequence of real numbers to converge to a limit.

We'll start by studying the real numbers in detail.  We'll identify their key properties, and see how from a relatively short list of these properties we can go on to deduce familiar facts such as that if \(ab = 0\) then \(a = 0\) or \(b = 0\), and if \(a\) is a real number then \(a^2 \geq 0\).

We'll go on to define convergence of a sequence of real numbers, and the related notion of a convergent series.  It was around two centuries ago that mathematicians identified these definitions, which fit with our intuition but also allow us to give rigorous proofs of the properties we expect of limits and infinite sums.

We'll explore some useful strategies for determining whether a series converges, and then conclude the course by studying power series, which are a specific type of infinite sum.  This will allow us to define and study familiar functions such as the exponential, sine and cosine functions.

This course leads on to Analysis II and Analysis III later in the year (of course), but will also feed into lots of later courses too.  In addition, it will give you an opportunity to develop useful 'habits of mind' regarding rigorous proof.