430 PROFESSOR YOUNG ON THE PRINCIPLE OF CONTINUITY. 



It will then only be necessary to suppose z to approach infinity as its limit; the connexion of 

 which limit, with the continuous set of values that it terminates, being preserved by the actual 

 exhibition of it; in its final state, or under the form co ; a symbol which, it will be observed, thus 

 spontaneoushi presents itself; and is not arbitrarily introduced to effect a purpose. 



Hence, when the limit I is reached through continuous ascending values of x, the extreme case 

 of the series (l) is 



and when it is reached through descending values, the extreme case of the series is 



I + — 



1 + — j - (l + — ] + he. ad inf. (4). 



And the values of these, as shown in the publication referred to, are respectively ^ and infinite. 



For any finite number of terms, these series do not differ sensibly from one another, nor from 



the neutral or independent series (2). But since we know that (I j = - , and I I + - J = e, 



it follows that, after a finite number of tern)s, the three series are totally distinct : and we thus see 

 that in such extreme cases as those we are now considering, it is not allowable, — as generally sup- 

 posed, — to neglect the terms infinitely remote from the commencement of the series : for it is only 

 in the infinitely remote region that the distinguishing peculiarities of the series become fully 

 ileveloped. And it is because of this, that in contemplating these extreme or limiting cases, differ- 

 ent orders of injinity become unavoidably forced upon our attention. Thus, in the infinitely remote 

 region of the series (3), it is obvious that there are places for the terms 



of which the numerical values are 



111 1 



-. -, -, — ,= 0. 



e e- e e=" 



And all these terms, as far as the zero-term, being significant, necessarily affect tlie numerical 

 i'X])ression for the sum of the whole ; and cannot be neglected with impunity in a correct estimate 

 of the value of the altogether boundless series (3). 



The theorems proposed by Cauchy, for testing the convergency of infinite series, do not apply 

 to the limiting cases, such as those here noticed. These theorems have in fact been the occasion of 

 error in the treatment of those cases; and it is one object of the present communication to invite 

 attention to this circumstance. 

 In discussing the series 



X .v' .r' .V' 



+ — H 1 + &c. ad inf. (5). 



12 3 4 •' 



Cauchv observes* that it will be convergent, or divergent, according as the numerical value of ,i' is 

 inferior, or superior to unity; but that when the limits .r = 1, w = - \ are actually reached, the 

 series will be divergent in the first case, and convergent in the second f. This is not a correct 

 account of what happens at the limits : if x ascend from an inferior numerical value (that is from 

 a fractional value, either positive or negative) up to iP = 1, or it = - 1, the limiting cases will he 

 tnnvergent, like all the preceding cases : but if the same limits be reached through descending 



• Cours d'Anatt/se, p. 153. f l>>'d., p. 155. 



