Theory and Application of Infinite Series (Dover Books on Mathematics) [Paperback]

Konrad Knopp (Author), Mathematics (Author)

Book Description

Series: Dover Books on Mathematics | Publication Date: March 1, 1990

Unusually clear and interesting classic 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, other topics). Exercises throughout. Ideal for self-study.

Editorial Reviews

Language Notes

Text: English
Original Language: German

Product Details

• Paperback: 563 pages
• Publisher: Dover Publications (March 1, 1990)
• Language: English
• ISBN-10: 0486661652
• ISBN-13: 978-0486661650
• Product Dimensions: 8.4 x 5.4 x 1.2 inches
• Shipping Weight: 1.3 pounds (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #437,701 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

15 of 18 people found the following review helpful:

5.0 out of 5 stars A Truly Marvelous Value, January 10, 2000

By 

Tony Aponick (Andover, MA) - See all my reviews

This review is from: Theory and Application of Infinite Series (Dover Books on Mathematics) (Paperback)

The last chapter on the Euler-MacLauren summation formula, and attendant interrelations among the Zeta function, Bernoulli Numbers and Bernoulli Polynomials is alone worth three times the price of this gem. Chock full of recipes and explanations of many of those little annoying points you don't understand fully. Do you REALLY understand what 'asymptotically equal to (~)' means? Heartily recommended!

25 of 32 people found the following review helpful:

4.0 out of 5 stars How to tell if you are a "formalist" --, February 15, 2002

By 

V. N. Dvornychenko (Rockville, MD) - See all my reviews
(REAL NAME)   

This review is from: Theory and Application of Infinite Series (Dover Books on Mathematics) (Paperback)

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!

6 of 8 people found the following review helpful:

5.0 out of 5 stars The best book on infinite series, October 9, 1998

By A Customer

This review is from: Theory and Application of Infinite Series (Dover Books on Mathematics) (Paperback)

Excellent book for consulting with lots of examples and problems. Very well written but with the problem of very old notation. Everything you need to know about series is in this book. Very good to use in problems seminars