This book is written primarily for undergraduates, though Part I may be judged by some teachers to be suitable for mathematicians during their last year at school. It includes the convergence theory that is commonly required for a university honours course in pure and applied mathematics, but excludes topics appropriate to post-graduate or to highly specialized courses of study. It has taken shape from sets of lectures I have given at various times during some fifteen years of university teaching.
The book develops the theory of convergence on the basis of two fundamental assumptions (one about upper bounds, one about irrational number as the limit of a sequence of rational numbers). With these assumptions the theory of convergence can be developed without appeal to the properties of Dedekind cuts. The 'real number' appears in the appendix, where the assumptions of the book are proved to be consequences of the definition of 'real number'.
The notation, or shorthand, used in the text is one that is familiar to the professed analyst and is a cornmonplace of the lecture-room. It is something of an experiment to employ it in a text-book, but its almost universal adoption in recent years by mathematical undergraduates at Oxford leads me to hope that it will prove acceptable. My own teaching experience is that students who use the notation acquire clear ideas of what they have to prove and of how they may prove it.
Of the details, few call for mention in the preface. The treatment of Tannery's theorem in Chapter XVI grew out of (i) Professor E H Neville's note in the Mathematical Gazette, vol. xv, p. 166, (ii) a remark once made to me by Professor Hardy, and (iii) my own work on a special series. I cannot resolve how much is due to each, but I am sure the chapter owes much both to Professor Neville and to Professor Hardy, and I gladly take this opportunity of acknowledging my indebtedness to them. The brief chapter on Fourier series will, I hope, prove useful in spite of its brevity and many omissions. The appendix contains just so much of the 'foundations of analysis' as is necessary to the justification of the assumptions made in the early chapters of the book. These 'foundations' are prefaced by a very brief historical sketch that tries to show why such a complex structure as a Dedekind out is necessary to the definition of 'number'.
All theorems are numbered. Some references to previous theorems are given in parentheses; if the reader can follow the proofs without consulting these references, so much the better. They are given so that readers may, if necessary, look up points they have forgotten: it is not intended that proofs in convergence theory should bristle with references to previous theorems in the manner of the old Euclid books. Though, of course, the order of proof is as important here as it is in the development of Euclidean geometry: we must not use A to prove B, and then use B to prove A.
The examples contain many questions set in university examinations and many questions taken from my own notes; of the latter, some are original and some are not. The majority are reasonably straightforward; hints for their solution are occasionally given. There are a few examples marked 'Harder', and the beginner is advised not to attempt them on a first reading.
Professor A L Dixon and Professor E T Copson have kindly read the proof sheets, and I am deeply grateful to them both for their helpful criticisms. Professor Copson has read and criticized all the text and has worked nearly all the examples. I wish to thank him most sincerely for this evidence of his friendship.
In conclusion, I should like to thank the staff of the Oxford University Press for their work on the book and for their unfailing courtesy towards me in all matters concerning it.
W. L. F.
16 November 1937.
Part I. A First Course in the Theory of Sequences and Series
I. Preliminary Discussion
II. Formal Definitions
III. Bounds: Monotonic Sequences
IV. Series Of Positive Terms
V. The Comparison Test; The Ratio Tests
VI. Theorems on Limits
VII. Alternating Series
Part II. The General Theory of Infinite Series
VIII. The General Convergence Principle
IX. Absolute and Non-Absolute Convergence
X. The Product of Two Series
XI. Uniform Convergence
XII. Binomial, Logarithmic, Exponential Expansions
XIII. Power Series
XIV. The Integral Test
XV. The Order Notation
XVI. Tannery's Theorem
XVII. Double Series
XVIII. Infinite Products
XIX. Theorems on Limits; Cesaro Sums
XX. Fourier Series
Real Numbers; Upper And Lower Bounds; Greatest And Least Limits
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