An Introduction to the Theory of Infinite Series - T.J.I'A. Bromwich

If you own only one book on infinite series, this is the book to own.  My copy is a second edition, first printed in 1928 by MacMillan - my copy is from 1949.  It has been reprinted by MacMillan and others since.  It's an easy read if you are familiar with Hardy's Pure Mathematics or other books at a similar level.  Coverage includes sequence, real, and complex series, power series and gamma functions.  The book contains an illuminating discussion of uniform convergence.  If you use series in your work as a mathematician or a physicist, this book should be on your bookshelf.