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Infinite Series

Isidore Isaac Hirschman
Publisher: 
Dover Publications
Publication Date: 
2015
Number of Pages: 
192
Format: 
Paperback
Price: 
14.95
ISBN: 
9780486789750
Category: 
Monograph
[Reviewed by
Allen Stenger
, on
03/28/2015
]

This is a concise introductory text on infinite series, that focuses on series of functions, especially power series and Fourier series. It is a 2014 unaltered reprint of the 1962 Holt, Rinehart & Winston publication.

The book develops everything from scratch, but because it is specialized, it is aimed at upper-division undergraduates. It has an enormous number of exercises, that gradually deepen from numerical exercises and convergence checking to various proofs and counterexamples.

The book does not use any complex analysis or the Lebesgue integral, but still manages to give a very broad view of the subject. The Fourier series portion is especially impressive; it is simplified by sticking to piecewise continuous functions, but hits all the major points of the theory, including summability. The power series portion sticks to the traditional real series development through Taylor series and the remainder formula, although there is also a chapter on real analytic functions.

This is a more advanced book than Bonar & Khoury’s Real Infinite Series, that deals only with series of numbers, and that book recommends Hirschman as a follow-on. The present book has largely the same coverage as Knopp’s Theory and Application of Infinite Series. Knopp goes into much greater depth, although it is slanted to series of numbers rather than functions.

Bottom line: an excellent introductory textbook, and valuable although not encyclopedic as a reference.


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.

  • Chapter 1. Tests for Convergence and Divergence
    • 1. Sequences
    • 1*. Limits
    • 2. Convergence
    • 3. Some Preliminary Results
    • 4. Integral Test
    • 4*. Error Estimates
    • 5. Comparison Test
    • 5*. Error Estimates Continued
    • 6. Relative Magnitudes
    • 7. Absolute Convergence
    • 8. Alternating Series
    • 9. Power Series
    • 10. Additional Exercises
  • Chapter 2. Taylor Series
    • 1 . Introduction
    • 2. Elementary Transformations
    • 3. The Remainder Formula
    • 4. The Remainder Formula (Continued)
  • Chapter 3. Fourier Series
    • 1. Classes of Functions
    • 2. Fourier Series
    • 3. Bessel's Inequality
    • 4. Dirichlet's Formula
    • 5. A Convergence Theorem
    • 6. Sine and Cosine Series
    • 7. Complex Fourier Series
  • Chapter 4. Uniform Convergence
    • 1. Sequences of Functions
    • 2. Uniform Convergence of Continuous Functions
    • 3. Miscellaneous Results
    • 4. Summation by Parts
  • Chapter 5. Rearrangements, Double Series, Summability
    • 1. Generalized Partial Sums
    • 2. Double Series
    • 3. Products
    • 4. Fubini's Theorem
    • 5. Summability
    • 6. Regularity
  • Chapter 4. Power Series and Real Analytic Functions
    • 1. Radius of Convergence
    • 2. Operations on Power Series
    • 3. End Point Behavior
    • 4. Real Analytic Functions
  • Chapter 7. Additional Topics in Fourier Series
    • 1. (C,1) Summability
    • 2. Uniform Continuity
    • 3. Parseval's Equality
    • 4. Convolution
  • Appendix
    • l. Set and Sequence Operations
    • 2. Continuous Functions
  • Index
  • Index of Symbols

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