152 Algebra. 



By Art. 25 the limit of i -\- x -\- x^ -\- . . . , as the number of 

 terms increases and as x approaches zero, equals i. Therefore, 



lim f.x-+;t"-h.r^+ . . . ^ = 

 and consequently 



lim K('.r^-.^"'4■-^■'+ . . . )=o. 

 That is to say, the right member of the inequality above can 

 be made as near zero as we please. Therefore since the left mem- 

 ber of the inequality is always numerically less than the right 

 member, the left member can be made to approach zero as near as 

 we please. Hence, 



lim ('A x+A x"-f A ^3+ . . . j = o. 

 That is hm (A,,-\-Ax-\-A,x'-{-Ax'-j- . . . ; = A^,. 



