INVESTIGATION OF A LIMIT. 



/' 1\* 



1 5. Limit of 1 1 H . The following theorem, which 

 V xj 



we proceed to demonstrate, is very important. When sc 



1 + - ) approaches a 

 x/ 



certain limit which lies betiveen 2 and 3. 



In the first place suppose x a positive whole number, = m 

 say ; we shall prove that the above expression continually in- 

 creases with m, but can never reach the value 3. Assuming 

 the Binomial Theorem for positive integral exponents, we have 



1.2 \mj 1.2.3 \m 



1.2. ..m 



which may be written 



l, T i\ l \ l ~"l ^""A 1 "^ 



WJ + 1 4 1.2 1.2.3 



m/ V m/ \ / , 



Similarly 



m+ , , 



Now in the last two series we see that their first and 

 second terms are equal, but the third term in (2) is greater 

 than the third term in (1) ; also the fourth term in (2) is 

 greater than the fourth term in (1), and so on; moreover 

 in (2) there is one term more than in (1). Hence 



/ i \ m+l . /: l V 



I 1 H is greater than 1 H 



V 1 + */ V m ' 



