ABSTRACT
One could argue that the concept of limit is themost fundamental one in analysis. Two of the most important operations in a first-year calculus course, the derivative and the integral, are in fact defined in terms of limits, even though many first-year calculus students willingly forget this is so. We begin this chapter by considering limits of sequences. The limit of a sequence of numbers, while the simplest kind of limit, is really just a special case of the limit of a sequence of vectors. In fact, real numbers can be considered geometrically as points in one-dimensional space, while vectors with k real components are just points in k-dimensional space. The special case of k = 2 corresponds to limits of sequences of points in R2 and to limits of sequences of points in C. Whether a sequence of real numbers, a sequence of real vectors, or a sequence of complex numbers has a well-defined limit is just a matter of determining whether the sequence of points is converging in some sense to a unique point in the associated space. This notion of convergence is one of the distance-related concepts referred to in the previous chapter, and it is common to all the spaces of interest to us. For this reason, we will again use the symbol X to denote any of the spaces R, Rk, or C in those cases where the results apply to all of them. After establishing the ideas underlying convergence of sequences in X, we develop the related notion of a series, whereby the terms of a sequence are added together. As we will see, whether a series converges to a well-defined sum depends on the behavior of its associated sequence of partial sums. While this definition of convergence for a series is both efficient and theoretically valuable, we will also develop tests for convergence that in many cases are easier to apply.
