﻿Exponential and Logarithmic Theorems. 45 



1 — — ) x&c, which must be an identical equation, its second 

 member being the factors of the first member ; or omitting the 



#3 7t' 2 7X* 



factor v, and putting 77=2, we get 1 -]~g3~ + l 2.3.45 z3 ~^ c ' 



(1 — *).(l-"~7"/( 1 -17/ X«fcc. ; •'. by equating the co-effi- 

 cients of the same powers of z, in the two members of the equa- 

 tion, using the notation in (a'), and the results in (e), we get 



1 1 1 



n 



6 



A = — 6=-\ 1 +4 + 9 + 16 +&c 'J' B = l20' C= -5040' ands0 



111 n* 111 



on; .'. l+^ + - +iI+ &c.=-g ,a=l+jj; +37+47 + &c. 



71 



4 - 11 



A'-2B=^, -R=l+^+3^+&c. = -(A3-3AB+3C)=^ 4 - 5 , 



and so on ; which agree with the results obtained by Enler at 

 p. 131 of his Analysis of Infinites: and by comparing the value 



of cos.i? given in (A) with its value given by (B'), we have 



l+rrr+^+&c.=^-, l+o7+^r+&c.=o7-, andsoon; as Eu- 



ler has given them at p. 132. We 



1) 



-SM 



3 y X . . . (l+jjTTi ' ' aUd by SU PP° si °§ M 



1W 1\ / i 



&c, or L.M = I l+.^ + 3 



finite, we have M=(l + l).( i l+^j^l+3J'\l+4/X&c. to in- 

 finity; .'.by taking the hyperbolic logarithm, we get L.M 



l +g + 5+i + &c " t0 inf -)4[ 1+ ^ + 3^ + &C - t0inf J + 

 If 1 1 1 / 1 1 



3 l +&+T 3 +&°' toinf - 



J(l+^+3\+&c.j4(l+^+3^ +&C -j +<fcc G' Wherethe 



quantity within the braces is evidently positive, we shall denote 



111 

 it by A, and we shall have l+2+3+4" f " &c * toinfin.=L.Mf A, 



.\ since M is infinite its logarithm is infinite, .'.the sum of the 



+ 



1 1 



