One very useful application of calculus involves the approximation of a function by a sequence of polynomials called Taylor polynomials. Before diving into the topic of Taylor polynomials, it will be wise to develop some of the general theory of series and their convergence. It is to this task that we devote the present chapter.
We begin by defining series and the notion of convergence, giving examples of series that do or do not converge. The primary method for determining whether or not a series converges is called the comparison test. When applied in the context of geometric series, it gives rise to another method called the ratio test. The integral (that most versatile creature) gives rise to the integral test. Alternating series come with their own test for convergence.
In conclusion, we anticipate our study of Taylor series by introducing power series, of which Taylor series are a special case. Power series contain a variable; whether or not they converge depends upon what value is substituted for the variable. As we will see, the values for which a particular Taylor series converges can be given quite a simple characterization.