Intégration et sommation : avec exercices / Hervé Lehning

The book, one of eight written for the students of mathematics at French universities, is divided into three parts: Théorie de l'intégration, Calculs d'intégrales, and Sommation. The first part presents an integration theory in which the integral of a piecewise continuous function f on [a,b] is defined as a limit of integrals of step functions uniformly approximating f. The Riemann theorem and the fundamental theorem of calculus are included as well as the definition and properties of improper integral and some criteria for convergence. The second part, after the list of well-known formulae and the two main methods (changing of variables and integration by parts), gives a limited survey of methods for calculating the integrals of some types of functions. The chapter "Calcul approché", also in this second part, is illustrated by computer programs and studies errors of various numerical methods. Most of the third part, after presenting the basic definitions and properties of numerical series, is devoted to the study of their convergence in analogy with the improper integrals. The last chapter gives some examples of calculating the sum of a series, both directly and by approximate methods, the latter being studied with a regard to the rate of convergence. The reviewer has successfully used the first two parts of the book for his course of mathematics at Ecole Nationale d'Ingénieurs de Gabès (Tunisia) and wishes to recommend it in view of the clarity of exposition, which stresses the key points, and the examples and exercises covering in a tiny volume much of the traditional material of elementary integration and infinite series theory. (Zentralblatt)

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