Combined Test of Convergence

The deductions of the Lagrange-Burmann, Taylor, and Stirling-Maclaurin (Chapter 23) and Teixeira and Laurent (Chapter 25) expansions of holomorphic functions, included indication of the regions of divergence and of convergence, but the behavior of the series on the boundary of convergence was not addressed. In the case of the Taylor series of a real function the convergence was not proved generally, and must be established on a case-by-case basis, using convergence tests. If a function is defined by a series, then the tests of convergence can be used to establish its properties, for example: (i) it is bounded if the series is conditionally convergent, and the order of the terms is not changed; (ii) it is continuous, differentiable, or integrable if the series is uniformly convergent, and consists respectively of continuous, differentiable, and integrable functions; (iii) the value of the function at a point can be obtained by summing the terms of the series in any order, if it is absolutely convergent. Next is presented a combined test of convergence (Section 29.1) that (i) applies to a wide class of series of functions commonly encountered in applications, including power and harmonic series; (ii) indicates the behavior of the series, that is, divergent, oscillating, or conditionally, absolutely, uniformly, or totally convergent (Chapter 21); (iii) applies to all points of the complex plane, including the whole of the boundary of convergence. The theorem (Section 29.1) combines the tests of D’Alembert (Section 29.3), Gauss (Section 29.5), and Abel-Dirichlet (Section 29.6) plus the Weierstrass K-test (Section 29.7), and is used to specify the behavior of series of functions at all points of the complex plane, for example, for the Gaussian hypergeometric series (Section 29.9). The establishment of the combined convergence test uses as intermediate steps alternative convergence criteria (Section 29.2), the comparison of convergence of series and integrals (Section 29.4) and the concepts of radius and exponent of a series (Section 29.8). The combined convergence test establishes the convergence properties at every point of the complex plane for simple series, for binomial (Sections 23.9 and 29.1), logarithmic (Sections 21.8 and 29.9), and Gaussian (Section 29.9), and generalized (Example 30.30) hypergeometric; it also applies to most series whose general term is of a not too “pathologic” form. It thereby indicates the extent to that can be used the solution of for example, a problem in terms of series.

The convergence of a series is addressed (Subsection 29.9.1) by a combined test applying inside, outside and at the boundary of convergence (Subsection 29.1.2), taking as first example the binomial series (Subsection 29.1.3).