IN REFERENXE. TO CERTAIN RESULTS OF ANALYSIS. 431 



values of x, the extreme cases will then, on the contrary, be divergent. The truth of this will 

 appear by writing these extreme cases with the proper symbol or indkation of continuity, intro- 

 duced, or rather preserved, as in the instances above. For we thus get the converging series 



and the diverging series 



That the former of these is convergent is obvious: and that the latter becomes divergent, in its 

 infinitely remote terms, will be seen from the following considerations : — 



As noticed above, fl + — I =e; so that, in the inlinitcly remote region, there occur the 



terms 



e e' e" e"' e^"' e^"' 

 03' 2x ' 3» ' w . es" 2».eo'" 3w . 03" 



e . e 



which evidently diverge after the term ,, and, in fact, after — . 



cc . ce eo 



Similar reasoning applied to 



1 + a; + '■Zx' + -l . 3,i' +2.3. 4.** + &c. ad inf. (6), 



another of the series considered by Caucliy, and which he affirms to be equal to I when *• becomes 

 zero, will show, that instead of 1, the value is infinite. For, writing the zero in the allowable 



form — , we find among the terms infinitely remote, the following: viz. 



2 . 3 .4 ... M 2 .3. 4 ... CO ... cc' 



00 ) ^, • &c. 



CO 05 



in which, as a' may exceed co in any ratio, the numerator may exceed the denominator in any 

 ratio ; so that the terms at length become infinitely great ; that is to say, the extreme case, 

 corresponding to a? = 0, is like all the other cases, divergent. 



1 he preceding reasonings, in which terms infinitely remote, and infinites of different orders, are 

 considered, may perhaps be regarded as too vague and subtil to justify an unhesitating recep- 

 tion of the conclusions to which they lead : and although they do not appear to me to be fairly 

 chargeable with this objection, yet I wish them to be regarded — less as demonstrations of the truth 

 of these conclusions, than as confirmations, supplied by the laws of analysis — when these are allowed 

 to have their full and unrestricted scope — of the general axiom which stands at the head of this 

 paper; and in virtue of which, if it be demonstrated, that an assigned analytical formula correctly 

 expresses the sum of an infinite series for all cases short of a cei tain extreme case — however closely 

 to this case we approach, — then we may safely infer that it equally, and as correctly, expresses 

 the sum in the extreme case also: a fact which is as necessarily true as any of the axioms of 

 Euclid ; and which I think can be questioned only by those who overlook the controlling influence 

 of the law of continuity over these terminal cases. It would be very wrong, in utter neglect of this 

 law, to confound the series 



V - 2' + :i' - 4'' + &c., 

 for instance, with wliat 



I* - 2',r + :i'a;' - 4-'ci'' + &c. 



becomes in the extreme case of .c <= I ; and thence to assert, as indeed has been done, that its sum is 

 Vol.. VIII. Paut IV. 3K 



