APPROXIMATIONS 9 



Then (1 - 2#)- 3 = {!+( 



-8) ( -2x) + 6( -2*) 2 + ( -10) ( - 



= 1 +6x +24tf 2 +80x* +2402^ -f 



3 



Expand (1 x)~^ to five terms. Here n = - 



Then (1 - a?) 1 ={!+(-)} "' 



234 



3 12 , 52 234 . 



1 H x H -- 2 H -- a? 3 H -- x* + 

 ^5 25 h 125 ^625 



7. Approximations. 



becomes (1 +a) =1 +na + M(> u" 1) a +n ( n - 1 H n " 2 ) 



. _ 



when a; = 1. 



If a is small compared with 1 and the value of n is kept reason- 

 ably small, between the limits + 2 and 2, the terms of the 

 expansion for (1 + a) n rapidly become smaller and smaller. 

 By taking the first two terms of this expansion, we obtain a first 

 approximation for (1 + a) n . 



The first approximations are : 



(1 + a) n *z 1 + na 



(1 a) n ^ I na 



(1 + a)~ n ~ 1 na 



(1 - a)- n 1 + na 



If a better result is required, the first three terms of the 

 expansion must be taken, and this gives rise to the second 

 approximations. 



