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**Theory and applications of infinite series.
6th ed.
(Theorie und Anwendung der unendlichen Reihen.)**
*(German)*
Zbl 0842.40001

Berlin: Springer-Verlag. xvi, 582 p. (1996).

Konrad Knopp’s classic text on the theory and applications of infinite series appeared in 1921 as Volume 2 in the distinguished Springer-Verlag series Grundlehren der mathematischen Wissenschaften. For three quarters of a century it has been a world standard, a favorite of students and professionals, also in its English and Polish translations. Now, eighteen years since the publication of the fifth edition we have the sixth German edition. Since changes from the previous edition appear to be minimal, it may be in order to review here very briefly the book’s structure. It is divided into three parts. Part one, Real numbers and sequences, takes the rational numbers as an ordered field as its basis, develops the real numbers as a complete ordered field and makes a thorough study of real number sequences. Part two, Foundations of the theory of infinite series, develops the basic convergence theorems, power series, the elementary functions, infinite products and summation in closed form. Part three, Extension of the theory, includes more advanced tests for convergence, series of functions, series with complex terms, divergent series and asymptotic expansions. Each of the 14 chapters closes with a set of exercises, 225 in all. As Wolfgang Walter writes in the Foreword: “Knopp writes with clarity and precision, he develops his topic without hurrying. He writes fluently and is never longwinded \(\dots\). His style is deft, even elegant, and its reading is a pleasure.” It is gratifying to see a new edition of Knopp’s book appear, and at a price which makes it affordable to students.

Reviewer: G.A.Heuer (Moorhead)

### MSC:

40-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to sequences, series, summability |