Double Series and Improper Double Integrals

Abstract

In this chapter, we shall develop the theory of double sequences, double series, and improper double integrals. Our treatment will be analogous to the treatment of sequences, series, and improper integrals of functions of one variable given in Chapter 9 of ACICARA. Much of this chapter can be read independently of the previous chapters of this book.

In the preamble to Chapter 2, we mentioned that the notion of sequences in ℝ, that is, functions from ℕ to ℝ, admits two generalizations in the setting of two variables: pairs of sequences and double sequences, that is, functions from ℕ to ℝ² and functions from ℕ² to ℝ. The former were discussed in Section 2.1 and we shall now take up a study of the latter. Thus, in Section 7.1 below, we outline the theory of double sequences and the associated notions of convergence, boundedness, monotonicity, etc. Double series and their convergence is discussed in Section 7.2. Various tests for determining the convergence or divergence of a double series are given in Section 7.3. In Section 7.4, double power series are treated as a special case of double series, and Taylor double series of infinitely differentiable functions are treated as a special case of double power series. We then turn, in Section 7.5, to a “continuous” analogue of double series, namely improper double integrals of functions defined on a set of the form [a,∞) × [c,∞), where a, c ∈ ∝. Tests for the convergence of an improper double integral are given in Section 7.6. Finally, in Section 7.7, the process of double integration is extended to functions defined on an unbounded subset of ℝ², and to unbounded functions defined on a bounded subset of ℝ².