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Theory and Application of Infinite Series (Dover Books on Mathematics), by Konrad Knopp, Mathematics
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This unusually clear and interesting classic offers a thorough and reliable treatment of an important branch of higher analysis. The work covers real numbers and sequences, foundations of the theory of infinite series, and development of the theory (series of valuable terms, Euler's summation formula, asymptotic expansions, and other topics). Exercises throughout. Ideal for self-study.
- Sales Rank: #615866 in Books
- Published on: 1990-03-01
- Released on: 1990-03-01
- Original language: English
- Number of items: 1
- Dimensions: 8.45" h x 1.16" w x 5.40" l, 1.28 pounds
- Binding: Paperback
- 592 pages
Language Notes
Text: English
Original Language: German
Most helpful customer reviews
0 of 0 people found the following review helpful.
For math purists only
By Tiptonville C.
I got bogged down in the introductory chapters. The author starts from the number 1 and wants to prove everything from there.
38 of 47 people found the following review helpful.
How to tell if you are a "formalist" --
By V. N. Dvornychenko
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!
4 of 4 people found the following review helpful.
Kindle edition flawed
By John W. Scott
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.
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