90 EXPANSION OF FUNCTIONS. 



where 6 is between and 1 ; 



(l-0} n x n+l _fx-0x\ n x 

 (l-0x) n+l '' = (l-t)x) 'I- Ox 



x - 0x* 



Tf -L i - A , . x0x , (x - 0x\ 



It x be less than unity, so also is - ~- , and 



1 0x \1 ray 



can be made as small as we please by taking n large enough. 

 Hence, if n be taken large enough, the remainder can be 

 made as small as we please. 



118. In the preceding Examples, we have been able to 

 write down the general term of the series, and the remainder 

 after n+ 1 terms. But if f(x) be a complicated function, the 

 expression for f n (x) will be generally too unwieldy for us to 

 employ. It is, therefore, not unusual to propose such ques- 

 tions as " expand e* log (1 + x), by Maclaurin's Theorem, as 

 far as 'the term involving x s ." Here we are not required to 

 ascertain the general term, or the remainder, or to shew when, 

 for the purpose of numerical computation, the remainder may 

 be neglected. We proceed thus : 



/{)-*** (!+) 



therefore /(O) = 0. 



By Ait. 80, 



therefore / (0) = 1 ; 



therefore /" (0) = 1 ; 



/'"<*>= 



therefore 



^ (x} 



therefore 



therefore / v (0) = 9. 



