8 INVESTIGATION OF A LIMIT. 



Therefore if we put m successively equal to 2, 3, 4_ &c. the 



f 1 \ m 

 expression f 1 H j continually increases. 



123 



But since 1 , 1 , 1 , ... are all positive and 



m m m 



all less than unity it follows that the series in (1) cannot be 

 greater than 



, 1 J^ 1 1_ 1 , . 



11.2 1.8.8 1.2.8.4 1.2...m" 



however great m may be. 



But the series in (3) is less than the following series, 

 which forms a geometrical progression, beginning at the 

 second term, 



f 1 + 2 + tf* 2 s + " + 2T* ' 

 that is, the series in (3) is less than 



Hence ( 1 4- } is less than 3, however great m may be. 

 V / 



/ 1 \ m 

 Since then the expression ( 1 H j continually increases 



with ???, but at the same time cannot exceed 3, there must 

 be some " limit" towards which it approaches as m is in- 

 creased indefinitely. We shall use the symbol e to denote 

 this limit, and shall hereafter shew how to calculate its 

 approximate value : we say approximate, for it will prove 

 to be an incommensurable number. See Art. 115. 



16. We might perhaps leave it to the student to convince 



himself that the limiting value of f 1 + J must be the same 



\ xi 



whether we attribute to a; a succession of integral or of 

 fractional values increasing without limit. But it may be 

 formally shewn thus. Whatever fractional value be ascribed 

 to x there must be two consecutive integers, say m and m + 1, 

 between which such fractional value lies. Suppose then 



