Through the three sections in this unit and the two sections in the next, we look at a few definitions and then study when series converge and diverge. We discover a number of tests that will help us figure out this information, including some very unusual-looking ones. The goal of this chapter is to allow us to use the information here when we try to express functions using certain infinite polynomials (we will call them power series). Knowing when a series converges or diverges will let us find out when our power series exist.
- Section 9.2: We begin with some book-keeping theorems about sequences. We want to know whether they behave the way that we would like. I doubt it's much of a spoiler to say that they do.
- Section 9.3: Series are a particular type of sequence - one that is created by adding the terms of a previously-given sequence. They form the backbone of the next two chapters and here we introduce two of the nicest ones.
- Section 9.4: These two tests begin the study of when series converge and diverge. The Divergence Test is weak, but very easy, while the Integral Test is quite powerful, but annoyingly difficult. Well, we need to begin somewhere...