J A Green
This book is intended primarily for students of science and engineering. Its aims are, first, to present the fundamental mathematical ideas which underlie the notion of a convergent series, and secondly to develop, as far as the small space allows, a body of technique and a familiarity with particular examples sufficient to make the reader feel at home with such applications of infinite series as he is likely to meet in his scientific studies.
I do not believe that these two aims are mutually antagonistic. It is true that a certain sophisticated skill is necessary for the construction of proofs of even quite elementary theorems involving, for example, the definition of the limit of a sequence, and that the acquisition of such skill would take more time than the non-specialist mathematician can spare. But this does not mean that either the fundamental definitions or the statements of the theorems cannot be clearly understood by the non-specialist; on the contrary, it is essential that they should be understood.
Accordingly it has been my policy to lay more emphasis on the illustration of basic ideas by numerical examples, than on formal proofs; the latter have often been relegated to small print, or omitted (such omissions are noted in the text). In particular the idea of convergence itself is directly involved in the practical problem of numerical calculation of the sum of a series, and I have devoted some space to this topic, traditionally neglected in elementary books on series.
It is a great pleasure to acknowledge my debt to my colleagues at Manchester, and especially to Dr W Ledermann, for their constructive comments at every stage.
J A GREEN
The University, Manchester