EXPANSION OF FUNCTIONS. 91 



a? 2x a 9ie 5 



Hence e*log (1 + )= + . + -^ + j-r+ .... 



L !_. L 



This may be verified by multiplying the expansion for e* 

 by that for log (1 + x). 



119. Methods of expansion of more or less rigour are 

 often adopted in special cases of which we will proceed to 

 give examples. We do not lay any stress upon them as 

 exact investigations, but they may serve as exercises in dif- 

 ferentiation. 



Expand tan" 1 .-*; in powers of x. 

 Assume ta,n~ l x = A + A l x + A z x' + ... + A n x n + ........ (1). 



Differentiate both sides with respect to x, 



then 2 = A 1 + 2A z x+...+nA n x n - 1 + ............ (2). 



But -^l-a^ + ^-z'+o: 8 - .............. ..... (3), 



by simple division, or by the binomial theorem. 



Equating coefficients of like powers of x in (2) and (3), 

 we have 



J 1 = l, A 3 =0, A = -i ^4=0,... 



and putting x = in (1), we get A = ; therefore 



* 3 * 5 x\ 

 tan l x = x + +... 



o O ( 



This example may also be easily treated by the rigorous 

 method already used in Arts. 11 4... 117. It appears from 

 Example 18, page 65, that the n tb differential coefficient of 

 tan" 1 ^ with respect to x is 



(-1)"-*|-K-1 . fn-rr ., \ 



^ '- = sin -- n tan x . 



