REPORT ON CERTAIN BRANCHES OF ANALYSIS, 2G9 



finite distance from the origin would be perfectly safe as far as 

 it does not influence the determination of the series from the 

 generating function, or the generating function from the series ; 

 and it is upon this principle that the practice is both founded 

 and justified. A few examples may make this reasoning more 

 plain. 



Let it be required to determine the function which generates 

 the series 



a + a X + a a:^ + aa^ + &c. (1.) 



Let s be taken to represent this function, and therefore 



s = a + ax-\-ax^-\-aa^ + &c. 



z= a -\- X [a + a X + a x^ -It a 3^ ->r &c.} 



=z a -{■ X S'. consequently 



a 



s = .j . 



\ — X 



If the arithmetical values of the terms of this series be con- 

 sidered, and if x be less than 1 , then , is the smn of the 



i — X 



series : in all other cases it is its generating function. 



We may consider, however, s (whether it expresses a sum or 

 a generating function) as identical with *,, ^g, ^g, &c., in the 

 several expressions 



s = a + X s^ 



s = a + a X -\- x"^ So 



s =^ a + a X -\- a x"^ -{■ x^ 



s = a + ax-\-ax^+ ...« .r™-' + x'"' s,„ : 

 for if the number of terms of the series s be expressed by n 

 and if n be infinite, we must consider *,, *2) *3) a ... Sm as abso- 

 lutely identical exTpvessions ; for otherwise we must consider an 

 infinite as possessing the properties of an absolute number, and 

 must cease to regard infinities with finite diflferences as iden- 

 tical quantities when compared with each other. It is for this 

 reason that we assume it as a principle that no regard must be 

 paid to terms at an infinite distance from the origin, whatever 

 their arithmetical values may be. 

 The sum of the series 



a — a + a — a-\- &c. 

 was assigned by Leibnitz, upon very singular metaphysical 

 considerations, to be — : the principle just stated would allow 

 us to put 



