Convergence of Sequences and Series
Welcome to Week 11.
In Lecture 21 on Wednesday, 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 version of) Lecture 21.
Welcome to Week 12, the last week of our semester.
In Lecture 22 on Thursday, we will look at another important application of the concept of convergence of a sequence, that is the fact that many important functions of mathematics can be represented as infinite power series. A power series in the variable x is like an “infinite polynomial”, it has a constant term, a x term, a x2 term, and so on. To say that the power series represents the function means that at least for some values of x, replacing x with th relevant value in the power series gives an infinite sum that converges to the value of the function at the relevant point. The power series representation may not resemble the definition of the function at all, and in a way this is the key point. Suppose we take a trigonometric function like cos – we know that for any (positive) real number t, cos(t) is the X-coordinate of the point that we would reach by starting at (1,0) and travelling a distance t counter clockwise along the unit circle. This is a definitive description of the meaning of cos(t), but it does not help us at all to determine the value of the cosine of 1 or 2 or 25 or any other number (except for some nice rational multiples of pi). But if we had an infinite series converging to cos(t), we could use that to at least approximate the value of cos(t), by truncating the series at some point and taking a finite sum. This principle is used a lot in practice, as are other features of series representation of functions. The concept has great importance in number theory and analysis as well as in modelling, numerical analysis and the study of differential equations.
In Lecture 22, we will look at the Maclaurin series representation of a function, which is the first step in the subject. We won’t prove any theorems, but we will see how our knowledge of differential calculus provides the basic tools needed to derive the coefficients in a power series representation.
Here is (an old version of) Lecture 22.
Relevant sections of the lecture notes this week are Sections 3.3 and 3.4 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.