Convergence of Sequences and Series
Welcome to Week 11.
We have two key ideas this week. In Lecture 20 on Wednesday, we will show that every convergent sequence must be bounded (this means that its set of terms is a bounded set). It is not true that every bounded sequence is convergent, and we have already seen examples to show this. However, if a bounded sequence has the additional property of being monotonic (this means that its successive terms either consistently increase, or consistently decrease), then it is convergent. This statement is known as the Monotone Convergence Theorem and it will be our main thing on Wednesday.
An example of a sequence that is not monotonic is any sequence that starts 1,2,3,2,1, . . . – this sequence increases for a few steps and then decreases, which is exactly what cannot happen in a monotonic sequence.
Here is (last year’s version of) Lecture 20.
Slides for Lecture 20, without annotation, and annotated.
In Lecture 21 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 (last year’s version of) Lecture 21.
Relevant sections of the lecture notes this week are Sections 3.2 and 3.3 in Chapter 3. There are more examples in the lecture notes than we will discuss in our lectures, and more detailed explanations in some places.
Supplementary video tutorial on Question 5, Sheet 61
A discussion of a problem similar to Question 5 on the current homework sheet Notes from this example.