Thomas J. Osler. 1. INTRODUCTION. This paper gives another proof for the remarkable simple

Size: px

Start display at page:

Download "Thomas J. Osler. 1. INTRODUCTION. This paper gives another proof for the remarkable simple"

Rachel Holly Potter
6 years ago
Views:

1 5/24/5 A PROOF OF THE CONTINUED FRACTION EXPANSION OF / Thomas J Oslr INTRODUCTION This ar givs aothr roof for th rmarkabl siml cotiud fractio = 3 5 / Hr is ay ositiv umbr W us th otatio x= [ a; a, a2, a3, ] for th cotiud fractio x= a a a2 a 3 Th i th covrgt of x= [ a; a, a2, a3, ] is th fiit cotiud fractio [ ] / = a ; a,, a (If th domiator of a covrgt is zro, w ca simly ski it i i i i th suc of covrgts) It is asy to s that / = a /, ad ( ) / = aa / a It is th atural to dfi = a, =, = aa = ad = a To fid th rmaiig covrgts w us th followig thorm (s Hardy ad Wright [4, Thorm 49, ag 3]):

2 2 Thorm : For = 2, 3, 4, it is tru that = a 2 ad = a 2 Olds [6] gav a xository roof of th cotiud fractio 2/ k 2/ k = [ k ;3 k,5 k,7 k, ] H th showd that for k = 2, it could b algbraically maiulatd ito = [2;,2,] = (Hr th bar otatio mas = [2;,2,,,4,,,6,, ]) H also idicatd that his rsult could b covrtd to a siml cotiud fractio for 2/k Th work of Olds is basd o th arlir work of Hrmit [5] Coh [] stramlid Olds s roof ito a short rstatio It is th uros of this ar to xtd Coh s argumt to th mor gral cas of / W ot that Eulr [2], [3] foud ( ) = 2/ = [;4 2] Our roof diffrs from othr roofs w hav s i that w obtai th rsult dirctly, without havig to covrt som othr cotiud fractio Prhas th most rmarkabl fatur of th roof is that w ar abl to xrss th rror ε k, which is giv by / / ε k = k k, xactly i trms of itgrals (Othr roofs listd i th rfrcs also shar this fatur) 2 THE CONTINUED FRACTION FOR / W ow rov that / = [;,,,3,,,5,,, 7,,,9, ] For th rmaidr of this ar ad rfr to th cotiud fractio for / From Thorm, w obtai =, 3 = 3 3 2; () = ((2 ) ), ((2 ) ) = ; (2) =, 3 2 = 3 3 (3)

3 3 for =,, 2, 3, Lt x ( x) A dx! =, x ( x) B dx! =, ad x ( x) C dx! = W will show that for =,, 2, 3,, A 3 / =, (4) 3 3 B 3 / =, (5) 3 3 ad C 3 2 / = (6) It is ot hard to s that as th right-had sids of (4), (5), ad (6) all aroach zro Thus oc w rov (4) - (6) (Thorm 3), w will hav show that th cotiud fractio xasio covrgs to th umbr / W bgi with a thorm rlatig th thr itgrals Thorm 2 For =,, 2, 3,, th followig ar tru: (a) A = B C, (b) ((2 ) ) B = A C, ad (c) C = B A

4 4 Proof It is asy to s that (c) is tru by xadig ( x ) x( x ) ( x ) = i th itgral dfiig C Nxt otic that d x ( x) x ( x) x ( x) x ( x) dx! ( )! ( )!! = Itgratig from to w gt = C / B / A /, ad thus (a) is rovd Fially, w start with d x ( x) x ( x ) ( ) x ( x) x ( x) dx! ( )!!! = which aftr som maiulatio rducs to, d x ( x) dx! = ( ) (2 ) x ( x) x ( x) x ( x) (!) ( )!! Itgratig from to w gt (2 ) C B = A, which rovs (c) Thorm 3 Euatios (4), (5), ad (6) ar tru for =,, 2, 3, Proof Our roof rocds by iductio o W lav it to th radr to show that (4), (5), ad (6) ar tru wh = W assum that (4) - (6) ar satisfid for = N W bgi by rovig (4) for = N Usig () w s that ( ) ( ) = / / / 3( N ) 3( N ) 3N 2 3N 2 3N 3N

5 5 By th iductio hyothsis i combiatio with (5) ad (6), w gt = C B Ivokig Thorm 2(a) w arriv at (4) / 3( N ) 3( N ) N N Nxt w stablish (5) for = N From (2) w ifr that ((2 3) )( ) ( ) = N / / / 3( N ) 3( N ) 3( N ) 3( N ) 3N 2 3N 2 Aalig to th idtity = A, (6), Thorm (2b), ad th / 3( N ) 3( N ) N iductio hyothsis, w s that (5) is tru Fially, w rov (6) for = N It follows from (3) that ( ) ( ) = / / / 3( N ) 2 3( N ) 2 (3N ) (3N ) (3N ) (3N ) Usig = B ad Thorm (2c), w s at oc that (6) holds This / (3N ) (3N ) N comlts th roof of th Thorm 3 I [] Coh suggsts th motivatio that may hav ld Hrmit to xami itgrals of th form m x x ( x ) dx wh h xamid th cotiud fractio for Rfrcs [] H Coh, A short roof of th siml cotiud fractio xasio of, Amr ath othly (to aar) [2] L Eulr, D fractioibus cotiuis dissrtatio, Comm Acad Sci Ptrool 9(744) 98-37; also i Ora Omia, sr, vol 4, Tubr, Lizig, 925, 67-25; Eglish traslatio by Wyma ad B Wyma, A ssay o cotiud fractios, ath Systms Thory 8 (985) [3] , Itroductio to Aalysis of th Ifiit, Book I (tras J D Blato), Srigr-Vrlag, Nw York, 988

6 6 [4] G H Hardy ad E Wright, A Itroductio to th Thory of Numbrs, Oxford Uivrsity Prss, Nw York, 98 [5] C Hrmit, Sur la foctio xotill, Comt Rd Acad Sci Paris 77 (873) 8-24, 74-79, , [6] C D Olds, Th siml cotiud fractio xasio of, Amr ath othly 77 (97) athmatics Dartmt, Rowa Uivrsity, Glassboro, NJ 828, Oslr@rowadu

Chapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series

Chapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris