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cr> CM CM BH EnHI UNIV.OF TORONTO BINDING 1 AX [NTRODUCTION To TIIK THEORY OF INFIMTK SKIMES MACMI-LLAN AND CO., LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO ATLANTA SAN FRANCISCO THE MACMILLAN CO. OF CANADA, LTD. TORONTO AN INTRODUCTION TO TIIK TIIKoRY OF INFINITE SERI1> BROMWH'H. T. .1. LA. M.A.. F.fc/8 KKI.I FORMKRLY PROFESSOR OF M ., MACMILLAN AN] LIMITED ST MAR UN'S STREET, LONDON 1908 ($>/? GLASGOW: PRINTED AT THE UNIVERSITY PR1 hV KOHKHT MA<1.KH08K AND CO. LTD. PREFACE, THIS hook is based on courses of l< Analysis given ;it (^m-i-n's Coll.-^.-. ( ialway. dm : Mi" LOUS I!MiM!M)7. Hut additions have naturally ln-,-n n. in preparing the manuvript fot press: in parti 1 hapter X I. ami th r part of t : been added, in selecting tin- subja r, 1 !ia\-.- attempt t> include proofs >!' all tliron-n, . in Priu^lM.-iin'- Irrationalzahlen ////'/ I\ wnendlicher itli th- -\<-rpti<.n oi' tli.-oi-rins n-latin^- to coiitinu'l tVad i. n-. In Chapter I. a preliminary account is 1' the n- j i limit ami ol' COTr I ha\e not in this clia- ath-iiipt.-.l to supply arithmetic proof- ,,f the fun-i thecrein^ (-..nctTnin-- the existence of limits, l.m bave allow-'-l their truth to rest on an appeal to the reader's intuition the hope that the diseu^ion may thus lr mai-- i, live to Ih-^inners. An arithnietie treatment will !> f"iin<l in Appendix J.. where Dedekiml's definition ..i irratinnal num; is adapted as fundamental: this metliod lea<U at the niom.tonic ]rim-ipl- of , mce (Art. 14!h. I'roni \\hicli e\i breme limitst ia deduced L60);itist] < easy t" .-taMish the ^-m-ral principle of In the remainder <,f the l,(...k fiv- n>- is made ,,f the n .1 and principles of the 1 >itl-ivntial and Integral Calculus: I 1 for BOme time l>em c.mvince.l that b uld not attempt to study Intinite Series in any detail until after they 1 . H.I. I.. 1'1>t. N17ot;ino.lnlHy'Jlli)t.Tf. Imt in ni:tny <>th-: : M have been >e by a systtnur i.uin and ininiinuin limits. vi PKEFACE. mastered the differentiation and integration of the simpler functions, and the geometrical meaning of these operations. The use of the Calculus has enabled me to shorten and simplify the discussion of various theorems (for instance, Arts. 11, 61, 62), and to include other theorems which must have been omitted otherwise (for instance, Arts. 45, 46, and the latter part of 83). It will be noticed that from Art. 11 onwards, free use is made of the equation although the limit of (1+1/V)" (from which this equation is commonly deduced) is not obtained until Art. 57. To avoid the appearance of reasoning in a circle, I have given in Appendix II. a treatment of the theory of the logarithm of a real number, starting from the equation X l The use of this definition of a logarithm goes back to Napier, but in modern teaching its advantages have been overlooked until comparatively recently. An arithmetic proof that the integral represents a definite number will be found in Art. 163, although this fact would naturally be treated as axiomatic when the subject is approached for the first time. In Chapter V. will be found an account of Pringsheim's theory of double series, which has not been easily accessible to English readers hitherto. The notion of uniform convergence usually presents diffi- culties to beginners; for this reason it has been explained at some length, and the definition has been illustrated by Osgood's graphical method. The use of Abel's and Dirichlet's names for the tests given in Art. 44 is not strictly historical, but i^ intended to emphasise the similarity between the tests for uniform convergence and for simple convergence (Arts. 19, 20). In obtaining the fundamental power-series and products constant reference is made to the principle of uniform convergence, and particularly to Tannery's theorems (Art. 49) ; the proofs are thus simplified and made more uniform than is otherwise possible. Considerable use is also made of Abel's