An Introduction to the Theory of Infinite Series (Classic Reprint)

This book is based on courses of lectures onE lementary Analysis given at Queen sC ollege, Galway, during each of the sessions 1902-1907. But additions have naturally been made in preparing the manuscript for press: in particular the whole of Chapter XL and the greater part of the Appendices have been added. In selecting the subject-matter, I have attempted to include proofs of all theorems stated in Pringsheim sarticle, I rrationalzafden und Konvergenz unencU icher Prozesse with the exception of theorems relating to continued fractions. In Chapter I. a preliminary account is given of the notions of a limit and of convergence. I have not in this chapter attempted to supply arithmetic proofs of the fundamental theorems concerning the existence of limits, but have allowed their truth to rest on an appeal to the readers intuition, in the hope that the discussion may thus be made more attractive to beginners. An arithmetic treatment will be found in Appendix I., whereD edekind sdefinition of irrational numbers is adopted as fundamental; this method leads at once to the monotonic principle of convergence (A rt. 149), from which the existence of extreme limits tis deduced (A rts. 5, 150); it is then easy to establish the general principle of convergence (A rt. 151). In the remainder of the book free use is made of the notation and principles of theD ifferential and Integral Calculus; I have for some time been convinced that beginners should not attempt to study Infinite Series in any detail until after they have Encyldopddie derM athemcU iacken WissenscJ iatenf Bd. I., A, 3and G, 3(pp. 47 and 1121). tN ot only here, but in many other places, the proofs and theorems have been made more concise by a systematic use of these maximum and minimum limits.
(Typographical errors above are due to OCR software and don't occur in the book.)

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