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.