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.