MA180/186/190 Calculus Semester 2. Week 10: Sequences and Convergence
Welcome to Week 10.
We will start our work on Chapter 3 this week. 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 numerial 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.
In Lecture 18, we will begin with some examples involving the concept of convergence, and then give it a precise meaning, at least for sequences of real numbers. Here is Lecture 18 (Wednesday April 21).
In Lecture 19 on Thurdsay, we will look in more detail at convergence of sequence, and in particular discuss the Monotone Convergence Theorem. Here is Lecture 19 (Thursday April 22).
Relevant sections of the lecture notes this week are Sections 3.1 and 3.2 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.
Weekly Problem 10
The weekly problems are just for fun. They have nothing much to do with our curriculum. Please send me an email if you have a solution that you would like to share with the class!