48 



tral series. Thus — although the contrary has often been 

 affirmed — we cannot legitimately infer from this theorem, 

 without the aid of an additional principle, that 



I - 1 + 1 — 1 + 1 — 1 + &c. = i 



For, as already shewn, the series within the brackets at (13) 

 is deficient by a quantity, which in this case is ± 1. Intro- 

 ducing this, (13) gives for s the ambiguous result i ± •§■ ; 

 that is, 1 or 0. The additional principle adverted to, and 

 which is absolutely essential to the received conclusion, is that 

 already stated at page 44 ; or, as Dr. Whewell briefly ex- 

 presses it, " that what is true up to the limit, is true at the 

 limit." 



The differential theorem, therefore, can never be employed 

 with success to sum either a divergent or a neutral series ; or 

 to convert either into a convergent series. 



There has been supposed to exist a perfect analogy between 

 1 — 1 + 1 — 1 + &c., as the limiting case of \ — g + g'^ — g^ + he, 

 and \-—2- + 3''-42 + &c., as the limiting case of 1 -2^ + 

 S^^r^ — 4-^^+ &c., and that, in consequence of this analogy, 

 we have as much right to affirm that P — 2'-'+ 3^— 4"^ + &c. 

 is accurately expressed by 0, the limiting case of (1 — g) 

 (1 -\-g)~^, the fraction which generates 1^ — 2^^+ 3'^^^ — 4^^''+ 

 &c., as that 1 — 1+1 — 1+ &c. is accurately expressed by 



i, the limiting case of — — — , the fraction which generates 



1 — r/ + g'~g^ + &c. But there is a total absence of analogy 

 between these tvvo instances : the series ^ — g -\- g^— g^-{- &c. 

 presents a series of convergent cases from ^ = 0, up to ^ = 1 ; 

 and whatever rule or formula enables us to find the summation 

 in all cases must necessarily enable us to find it in the extreme 

 positive limits and 1 ; for no values, short of those limits, 

 can be the first and last of the admissible cases. But this rule 

 or formula of summation, whatever it be, is constructed con- 

 formably to certain hypotheses ; viz. that the convergent 



