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Theory and Application of Infinite Series

Konrad Knopp
Dover Publications
Publication Date: 
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This is a comprehensive reference on infinite series. It comes at the subject from a function-theoretic point of view, that is: what functions can be represented by infinite series (particularly power series and Fourier series, but also some more specialized ones), and what can we infer about a function from its series? It starts from the beginning, and develops and defines the concept of convergence and includes a lot of material on convergence tests, but it is primarily about functions. There is also a moderate amount of information about transformation of series to increase the speed of convergence, and numerous examples of numerical calculation of function values from series.

The book was first published in German in 1921 and went through several revisions, and there were two English editions. The present volume is a 1990 Dover reprint of the 1951 English edition from Blackie & Son, which is based on the 1947 German edition. The book has aged well, and most of the material here is still valuable today. It omits coverage of some series that have become important today, such as the hypergeometric series. There are a few spots where the language has changed, mostly in the summability sections, but this causes little problem in following the narrative. There is no computer coverage or discussion of round-off error, and all the numerical examples assume hand calculation and are worked to about a dozen decimal places.

The most valuable part of the book is the examples and exercises. These are numerous and interesting, and appear at the end of each section and of each chapter. They nearly always ask for facts or proofs about particular series or functions, rather than to prove general theorems that were not covered in the body.

I don’t know of any modern books with the same coverage as this one. A near-contemporary is Bromwich’s 1926 An Introduction to the Theory of Infinite Series, that covers much the same topics and to the same level of detail. A good modern book is Bonar & Khoury’s 2006 undergraduate book Real Infinite Series, although that is much more limited and is concerned mostly with convergence tests and to some extent with identifying and evaluating series. Books on real analysis, complex analysis, and Fourier analysis often have good sections on the properties of the corresponding series. Summability tends to be studied in the context of Fourier series, but is also the subject of a whole book by G. H Hardy, the 1949 Divergent Series.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

  • Introduction
  • Part I. Real numbers and sequences.
    • Chapter I. Principles of the theory of real numbers
      • 1. The system of rational numbers and its gaps
      • 2. Sequences of rational numbers
      • 3. Irrational numbers
      • 4. Completeness and uniqueness of the system of real numbers
      • 5. Radix fractions and the Dedekind section
      • Exercises on Chapter I (1-8)
    • Chapter II. Sequences of real numbers.
      • 6. Arbitrary sequences and arbitrary null sequences
      • 7. Powers, roots, and logarithms. Special null sequences
      • 8. Convergent sequences
      • 9. The two main criteria
      • 10. Limiting points and upper and lower limits
      • 11. Infinite series, infinite products, and infinite continued fractions
      • Exercises on Chapter II (9-33)
  • Part II. Foundations of the theory of infinite series.
    • Chapter III. Series of positive terms.
      • 12. The first principal criterion and the two comparison tests
      • 13. The root test and the ratio test
      • 14. Series of positive, monotone decreasing terms
      • Exercises on Chapter III (34-44)
    • Chapter IV. Series of arbitrary terms.
      • 15. The second principal criterion and the algebra of convergent series
      • 16. Absolute convergence. Derangement of series
      • 17. Multiplication of infinite series
      • Exercises on Chapter IV (45-63)
    • Chapter V. Power series.
      • 18. The radius of convergence
      • 19. Functions of a real variable
      • 20. Principal properties of functions represented by power series
      • 21. The algebra of power series
      • Exercises on Chapter V (64-73)
    • Chapter VI. The expansions of the so-called elementary functions.
      • 22. The rational functions
      • 23. The exponential function
      • 24. The trigonometrical functions
      • 25. The binomial series
      • 26. The logarithmic series
      • 27. The cyclometrical functions
      • Exercises on Chapter VI (74-84)
    • Chapter VII. Infinite products.
      • 28. Products with positive terms
      • 29. Products with arbitrary terms. Absolute convergence
      • 30. Connection between series and products. Conditional and unconditional convergence
      • Exercises on Chapter VII (85-99)
    • Chapter VIII. Closed and numerical expressions for the sums of series.
      • 31. Statement of the problem
      • 32. Evaluation of the sum of a series by means of a closed expression
      • 33. Transformation of series
      • 34. Numerical evaluations
      • 35. Applications of the transformation of series to numerical evaluations
      • Exercises on Chapter VIII (100-132)
  • Part III. Development of the theory.
    • Chapter IX. Series of positive terms.
      • 36. Detailed study of the two comparison tests
      • 37. The logarithmic scales
      • 38. Special comparison tests of the second kind
      • 39. Theorems of Abel, Dini, and Pringsheim, and their application to a fresh deduction of the logarithmic scale of comparison tests
      • 40. Series of monotonely diminishing positive terms
      • 41. General remarks on the theory of the convergence and divergence of series of positive terms
      • 42. Systematization of the general theory of convergence
      • Exercises on Chapter IX (133-141)
    • Chapter X. Series of arbitrary terms.
      • 43. Tests of convergence for series of arbitrary terms
      • 44. Rearrangement of conditionally convergent series
      • 45. Multiplication of conditionally convergent series
      • Exercises on Chapter X (142-l53)
    • Chapter XI. Series of variable terms (Sequences of functions).
      • 46. Uniform convergence
      • 47. Passage to the limit term by term
      • 48. Tests of uniform convergence
      • 49. Fourier series
        • A. Euler's formulae
        • B. Dirichlet's integral
        • C. Conditions of convergence
      • 50. Applications of the theory of Fourier series
      • 51. Products with variable terms
      • Exercises on Chapter XI (154-173)
    • Chapter XII. Series of complex terms
      • 52. Complex numbers and sequences
      • 53. Series of complex terms
      • 54. Power series. Analytic functions
      • 55. The elementary analytic functions
        • I. Rational functions
        • II. The exponential function
        • III. The functions cos z and sin z
        • IV. The functions cot z and tan z
        • V. The logarithmic series
        • VI. The inverse sine series
        • VII. The inverse tangent series
        • VIII The binomial series
      • 56. Series of variable terms. Uniform convergence. Weierstrass' theorem on double series
      • 57. Products with complex terms
      • 58. Special classes of series of analytic functions
        • A. Dirichlet's series
        • B. Faculty series
        • C. Lambert's series
      • Exercises on Chapter XII (174-199)
    • Chapter XIII. Divergent series.
      • 59. General remarks on divergent series and the processes of limitation
      • 60. The C- and H-processes
      • 61. Application of C1-summation to the theory of Fourier series
      • 62. The A-process
      • 63. The E-process
      • Exercises on Chapter XIII (200-216)
    • Chapter XIV. Euler's summation formula and asymptotic expansions.
      • 64. Euler's summation formula
        • A. The summation formula
        • B. Applications
        • C. The evaluation of remainders
      • 65. Asymptotic series
      • 66. Special cases of asymptotic expansions
        • A. Examples of the expansion problem
        • B. Examples of the summation problem
      • Exercises on Chapter XIV (217-225)
  • Bibliography
  • Name and subject index