Theory and Application of Infinite Series (Dover Books on Mathematics) by Konrad Knopp (PDF)

Description

This classic work, written in a clear and interesting style, with many exercises, offers a thorough and reliable treatment of an important branch of higher analysis. It lends itself well to use in course work; however, because of its consistent clear illustrations of theoretical difficulties, the book is also ideal for self-study.Since all higher analysis depends on the theory of numbers, Professor Knopp (formerly Professor of Mathematics, University of Tübingen) begins with an introduction to the theory of real numbers, an indispensable foundation for what is to come. This introduction is followed by an extensive account of the theory of sequences and the actual theory of infinite series. The latter is covered in two stages: (1) the classical theory (2) later developments of the 19th century.Carefully selected exercises have been included throughout, emphasizing applications of the theory, rather than purely theoretical considerations.Aimed at students already acquainted with the elements of differential and integral calculus, this work grew out of the author’s lectures and course work at the universities of Berlin and Königsberg. This pedagogical background helped him achieve a work of utmost clarity and precision — one that belongs in the library of every serious mathematician or student of higher analysis.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Anything to do with “infinity” is fascinating. Much of the history of mathematics has been a duel between those who see “infinity” as a delusion and impediment to progress, and those who see it as the greatest tool in the mathematician’s toolbox. Infinite series, which may be loosely defined as sums of an infinite number of terms (numbers), take on some of this fascination. Although this book will appeal mainly to the professional mathematician, there is enough historical and elementary material to profit many college students- and possibly even some high school students.Professional mathematician will find this book useful for filling in gaps left by topics not covered in traditional courses. An example is the detailed discussion of Euler’s summation formula, which goes far beyond the simplified form usually encountered in textbooks. Another fascinating topic covered is divergent series, and methods by which meaningful sums can be assigned to these. There is something counterintuitive — and, frankly, mind-boggling — about many of these results.Mathematicians can be put into several categories: 1) applied-mathematicians/computer-scientists/engineers concerned with solving practical problems, 2) those concerned with pedagogy and the history of mathematics, 3) epistemology and rigorous proofs, and 4) formalists. The fourth category, formalists, is difficult to define, but may be described as those that emphasize obtaining new results through formal (technical) manipulations, without undue concern regarding the meaning of the intermediate steps. The greatest exponents of this art were Euler and Ramanujan, though Fourier, Dirac and Heaviside are also solid members of this camp.I take this digression because I feel that this book mainly appeals to the fourth type of mathematician. Although there are some general results in the theory of infinite series, any competent mathematicians can, in a few minutes, write a dozen infinite series which defy summation. As an example, the series associated with the Riemann zeta function of EVEN arguments were first summed by Euler. The sums arising from ODD arguments have defied summation to this day. Why this should be so is intriguing, but unknown. Incidentally, Euler’s method of summation will make a “rigorists” hair stand upon ends. But he got the job done!

⭐This book is one of the more dense math books I have read. In the first 10 pages it asks “Has anyone really beheld the square root of 2? Even if we estimate to the first one million digits, squaring it will not yield 2”. It then goes though the fundamentals of number theory and dives in to the complex and rich world of infinite series’. You will probably have to do some backtracking within the book and on the internet to understand a lot of the finer points in the book. It makes you work, but is worth it!

⭐This is a comprehensive book on the 19th century theory of infinite series. It is useful mostly for its good examples and its many references to the original writings of mathematicians such as Cauchy, Abel, Dirichlet, Jordan, du Bois-Reymond, and Dini. There is a chapter on summability methods for divergent series (Chapter XIII) and a chapter on asymptotic series (Chapter XIV).This book is similar to Bromwich’s “An Introduction to the Theory of Infinite Series”.

⭐like4 it,knowledge

⭐May not be the most modern text on the subject, but it is written clearly and carefully explains the subject. The introduction chapter is a very useful refresher for non- mathematicians. Overall it provides a good treatment of subject, recommended for self study provided you have a reasonable knowledge of calculus and complex numbers (for the later chapters.

⭐This is a classic book written by Knopp who shows a complete mastery of the subject. The construction of real numbers is unique and he uses nests to define them. If ever you wanted information on series then this book is a bible on that topic. Any student of mathematics must have this in his collection. The translator has done an excellent job.Dr. Knopp also has a series of books on the theory of complex functions and are highly recommended.

⭐The Kindle edition is a flawed copy of the paper edition. I only have two examples, but being a math text where every symbol counts, that’s enough for me to abandon the electronic version in favor of the paper version.Example 1: Page 6 in the print book compares sets named M and N, but on the Kindle these are both referred to as M.Example 2: Reference numbers set aside in the margin of the paper edition get interspersed throughout the Kindle edition. This is confusing as they sometimes come in the middle of an ordered list.I encountered both problems very early, so there may be many more problems. The text is already a challenge for me, so I don’t need additional, unnecessary problems to sort out. I do not recommend the Kindle version.

⭐This is a very fantastic book on infinite series. The coverage of subjects in the book is very comprehensive and goes way beyond a simple introduction to infinite series. Reading this book though, I would recommend teaching yourself Real/Complex Analysis along with this as it will give you greater insights. Overall though I am pleased with this book, and the price cannot be beat.

⭐I’m working through the kindle version of this text which is a well written undergraduate level treatment of sequences and have rapidly encountered numerous errors, such as less than signs instead of greater than signs etc. Whilst in some texts these could be overlooked, in an analytical text such as this they represent a significant problem.Also, despite having run to 4 editions, there are no solutions for any exercises in the text. I have little patience for maths texts that don’t offer worked solutions (or in this case any solutions at all).Nevertheless I’m continuing to use it as it seems to be a fairly comprehensive coverage of sequences. Would probably be better to stick with the print-version until the Kindle version is improved though.

⭐A classic in the field and quite complete. Cannot ask for better exposition on the theory and application of infinite series. Very pleased that I was able to get that book.

⭐The characters are blurred and very small. They are not impressed but photocopied. Very cumbersome to read.

⭐Excellent

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