Welcome to Week 10. The theme this week is to give a precise meaning to the very important concept of convergence of a sequence of real numbers. We will also discuss some other properties of sequences, such as boundedness and monotonicity. We will prove the Monotone Convergence Theorem, which states that the properties of boundedness and montonicity in combination are enough to guarantee that a sequence is convergent – along with the Axiom of Completeness in the real numbers.
Here is (an old video version of) Lecture 19
Slides from Lecture 19, annotated and without annotation.
In Lecture 20 on Thursday, we will look at how we can adapt the idea of convergence for sequences to define a concept of convergence for infinite series or infinite sums. Convergent sequences have many important roles in different areas of mathematics. One that we can see relatively quickly is that the concept and definition of convergence for sequences (which are just lists of numbers) allows us to define (and sometimes test) a criterion for when all the terms of an infinite sequence can be added together to produce a numerical value. A sum of this type is called an infinite series. Associated to every infinite series is its sequence of partial sums, which consists of the first term, the sum of the first two, the sum of the first three, and so on. If this sequence converges, the series is said to converge. We will look at this idea on Thursday, and consider how it applies in certain cases, such as geometric series and the important example of the harmonic series.
Here is (an old video version of) Lecture 20.
Slides for Lecture 20, without annotation, and annotated.
The relevant section of the lecture notes for this week is Section 3.2.