EXPANSION OF FUNCTIONS. 89 



In Arts. 115 and 116, the student will see that the last 

 term can be made as small as we please, whatever be the 

 value of x, if r n be taken large enough. 



117. Let /(o?)=log(l+aO; 



therefore /' (o>) = ^ and /' (0) = 1, 



and/n (o) = ( ~ 



therefore, by Art. 95, 



11-1 

 y 



mj 



T (n 4- 1) (1 + 6x} n " 

 In this series, if we suppose x positive and not greater 



/ y> Cn+1 



than unity, then, as ( ^- ) can not be greater than unity, 



\1 + uX/ 



f l)"" 1 ^; 



the error we commit, if we stop at the term , is 



n 



not greater than -; that is, the error can be made as 



small as we please by increasing n sufficiently* 

 If we change the sign of x, we have 



a) n+1 



log (I-*) =-*--- ---...---.y- 



which does not give a very convenient form to the remainder. 

 But by Art. 110, we may also write 



\_ v? v?_ x n (l-0) n x n+1 



~*~~~-~ ~-' ' 



