REPORT ON CERTAIN BRANCHES OF ANALYSIS. 277 



Such limitations will be conveyed by the introduction of signs 

 of affection, of signs of transition, or of signs of discontinuity, 

 which may be involved either implicitly or explicitly. It is for 

 such reasons that all those signs must be considered in the 

 interpretation of algebraical formulae, and their occurrence will 

 at once suggest the necessity of such an examination of the 

 circumstances of their introduction as may be required for their 

 correct explanation*. 



We thus recognise two classes of diverging series, which are 

 distinct in their origin and in their representation. The first 

 may be considered as involving the symbol or sign co implicitly, 

 and as capable, therefore, of the same interpretation as we give 

 to the sign when it presents itself explicitly. The second re- 

 presents finite magnitudes, which in their existing form are 

 incapable of calculation by the aggregation of any number of 

 their terms. Such series are in many cases capable of trans- 

 formations of form, which convert them into equivalent con- 

 verging series ; and in some cases, where such a transformation 

 is not practicable, or is not effected, the approximate values of 

 the generating functions may be determined, from indirect con- 

 siderations, supplied by very various expedients. 



The well known transformation of the series 



ax — bx^-'rca^ — dx^ + ex^— fx^ + &c., 

 which Euler has given f, into the equivalent series 



J a + T^— — vQ .A^a — -rr— — r-. . J^ « + &c. 



\ + x {l + xf "^ ' {\ + xf" '^ (1+^)^ 



would be competent to convert a great number of divergent 

 series of the second class into equivalent convergent series, or 

 into such as would become so. In this manner the Leibnitzian 

 series 



1 - 1 + 1 - 1 + &c. 



may be shown to be equal to -^. The series 



1 - 3 + 6 - 10 + 15 - 21 + &c. 



* The essential character of arithmetical division is that the quotient should 

 approximate continual]}' to its true value, and that the terms of the quotient 

 which are introduced by each successive operation should be less and less con- 



-tinually. In the formation, therefore, of the quotient of r and r 



■' ^ a — b a + b 



the analogy between the arithmetical and algebraical operation would cease to 



exist, unless a was greater than b, or unless the several terms h> the quotient 



■went on diminishing continually. 



+ Irutkutiones Calculi Differenlialh, Pars posterior, cap. i. 



