a' 



■••V. Ik 



1821.] a Method of applying Maclaurin's Theorem. W&l. 



By substituting for A, B, C, &c. we have 

 log. (a- z) = I. a - (i + ^. + ^-^^ + Jl. + &c.) 



Example 5. — Expand into a series the log. f ) log. of base 



unity. . - s ' . 



jf =- 7T7 =- G +/.-)= B +/. ^ .-.B = -i. 

 f^ = c7h? = ^- +/..- = 2C +/„.... C = ±, 

 f^=-Fl^=- (I +/'..-) = 2. 3D +/„...■.© 



1_ 



&C , 



Hence, and by substitution, we have 



Example 6. — Expand into a series the log. [ -—- ), log. of base 

 unity. 



« = '(r^.) = ^ (j) -^f- = A +/- ••• A = ; (i) ^^^ 



^ — r- = - + /i x; = B + /i « .•. B = -, r 



U = (^3 = !. +/.U . = 2 . 3 D +/„, . .-. D = 3L, 



&c 



Hence, and by substituting in the problem, we have 



Example 7. — Expand into a series a*, log. of base unity. -\ 

 u = a' = {1 + (a - l)}^= 1 + fx=:A -{-fx .-. A == I, 



~-= /a{l +(«-!)}* = /«+/ia;=B +f,x.\B = I .a, 



^^ = (/ . «)^{1 + (a - l)}^= (/ . «)^ +/,:r = 2 C +/, « 

 . p __ a. a)* 



7^ = (^ fl)^ {1 + (a - l)r = il . «)^ +/ni ^ = 2 . 3 ^ -h 



/.„.... D=|;4', 



&c. 



