## Sequences and Convergence

Welcome to Week 9.

We will finish working on Chapter 2 this week.

In Wednesday’s lecture, we will finish the discussuion of the Axiom of Completeness, and start our work on Chapter 3. The theme is convergence of sequences and series. We have already seen that an irrational number (for example) can be the limit of a sequence of rational numbers, we can think of π for instance as the limit of its successive decimal approximations. In this chapter we will give a precise meaning to the concept of convergence of a sequence, and use it to consider the question of whether the sum of infinitely many numbers can ever have a numerical value, and even whether a function can be represented as an infinite sum. Answers to these questions have very important applications to practical problems of approximating quantities that cannot be computed precisely, and even for determining values of everyday functions like the trigonometric ones.

Slides for Week 9.

Here are old video versions of Lectures 17 and 18. The version of Lecture 17 here is not quite the same as this year’s in-person class, it looks specifically at Question 2 on the exam.

Slides for Lecture 18, without annotation, and annotated.