ABSTRACT
The questions of convergence arise for infinite processes such as (1) series including power series (Chapters I.23, I.25 and I.27 and Section 1.1) or series of fractions (Sections 1.2 and 1.3) for functions analytic except for poles or essential singularities; (2) infinite products for functions with an infinite number of zeros that meet the Mittag-Leffler (Weierstrass) theorems [Sections 1.4 and 1.5 (Section 9.2)]; (3) nonterminating continued fractions (Sections 1.5 through 1.9) including recurrent and nonrecurrent (Section 9.3); and (4) improper integrals of the first (second) kind that is (Section I.17.1) with one or two infinite limits (an integral singular at a point within or at one end of the path of integration). A combined test of convergence has been derived (Chapter I.29) that applies at all points of the complex plane for a wide class of series of functions. The combined convergence test can also be applied to other infinite processes by relating them to series, namely, (1) an improper integral of the first or second kind of a monotonic function has lower and upper bounds that are specified by series and thus can be used to establish convergence (Section 9.1); (2) to each infinite product a series is associated that converges, diverges, and oscillates in the same conditions (Section 9.2); and (3) the convergence of a continued fraction can be established by considering some associated series (Section 9.3).
