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An Introduction to the Theory of Infinite Series

Thomas J. Bromwich
American Mathematical Society/Chelsea
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a thorough and still up-to-date introduction to infinite series. The present work is a 1991 AMS Chelsea unaltered reprint of the 1926 second edition from Macmillan.

The book is slanted towards mathematical physics, though without giving any applications there, and has a lot of material on differential equations, Fourier series, and Bessel functions. The book is especially good at counterexamples, and includes many of these to warn against pitfalls in reasoning and to show that all the hypotheses of the theorems are really needed. One especially nice feature is the use of Tannery’s theorem, on interchanging limit and summation, throughout the book. About a quarter of the book is in three appendices on background and related material not in the mainstream of the book.

Bromwich’s book was published at about the same time as Knopp’s Theory and Application of Infinite Series, and they have very similar tables of contents. Bromwich does not go into as much depth (it is more truly an introduction than Knopp, which is more of a reference), but Bromwich does have much harder and more numerous exercises. A good modern introduction to infinite series (though much more limited than either of these) is Bonar & Khoury’s Real Infinite 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.

  • Sequences and limits
  • Series of positive terms
  • Series in general
  • Absolute convergence
  • Double series
  • Infinite products
  • Series of variable terms
  • Power series
  • Special power series
  • Trigonometrical formulae
  • Complex series and products
  • Special complex series and functions
  • Non-convergent series
  • Asymptotic series
  • Trigonometrical series
  • Appendix I. Arithmetic theory of irrational numbers and limits
  • Appendix II. Definitions of the logarithmic and exponential functions
  • Appendix III. Some theorems on infinite integrals and gamma-functions
  • Miscellaneous examples
  • Index of special integrals, products, and series
  • General index