INVESTIGATION OF A LIMIT. 13 



If /3 and //. are positive quantities, and yu-/3 less than unity, 

 (1 +BY is less than =-3 . .. (2). 



1 f-L/3 



To establish these inequalities we shall use .the known 

 theorem that the arithmetical mean of any number of posi- 

 tive quantities is greater than the geometrical mean ; see 

 Algebra, Chapter LI. 



nr\ 



Let \ = ~ , where p and q are positive integers. Take p 

 quantities, q of which are equal to 1 + - /3, and p q equal to 



unity. Then their arithmetical mean is 



P 



f r> \ 3 - 

 that is 1+/3; their geometrical mean is ll + -{i\ p . The 



p 

 former is the greater; and therefore (1 + /3) 2 is greater than 



1 +& Thus (1) is established. 

 2 



O * 



Let /* = - , and ^u/3 = - , where r, s and t are positive in- 



t t 



ty* 



tcgers ; thus /3 = - . Take s + t quantities, s of which are 



S 

 7* 7" 



equal to 1 + - , and t equal to 1 . Then their arithmeti- 

 s t 



cal mean is\- - - - , that is unity; their geome- 



^v S ~T t 



(f f\ f r\'] t+e 



trical mean is -U 1.+ - ) ( 1 If- . The former is the greater ; 

 (V 6> /V / J 



f T\' f T\* 



therefore (l + -](l ) is less than unity; and therefore 



\ sj \ tj 

 >_ 



(r\ * 1 



1 + - is less than - . Thus (2) is established. 

 *J , r 



