Uniform Convergence of Sequences of Functions

Abstract

In this chapter we consider sequences and series of real-valued functions and develop uniform convergence tests, which provide ways of determining quickly whether certain sequences and infinite series have limit functions. Our particular emphasis in Section 9.1 is to present the definitions and simple examples of pointwise and uniform convergence of sequences. In addition, we present characterizations for interchanging limit and integration signs in sequences of functions. In Section 9.2, we discuss a characterization for interchanging limit and integration signs, and interchange of limit and differentiation signs for uniform convergence of sequences and series of functions. At the end of the section, we also include some foundations for the study of summability of series, which is an attempt to attach a value to a series that may not converge, thereby generalizing the concept of the sum of a convergent series. Finally, we also discuss the Abel summability of series. At the end of Section 9.2, we state and prove an important result due to Weierstrass, which in a simple form states that “any continuous function on [a, b] can be uniformly approximated by polynomials.”