432 



PROFESSOR YOUNG, ON THE PRINCIPLE OF CONTINUITY, 



zero, when in reality the sum is ± co . The erroneous sums assigned to divergent series, will bt 

 found in many other instances besides this, to belong, not to the independent series themselves, 

 but to the extreme cases of certain general forms. Yet the errors adverted to, and which formed 

 the subject of a communication submitted to the British Association in 18i4*, are not always of this 

 character: the value •596347..., for instance, assigned by Euler, and many succeeding writers, to the 

 series 



I-I+2-2.3+2.3.4.- &C., 



neither belongs to this series nor yet to the extreme case of the general series (6'), in whicli 

 «• = 1 — — ; since we have seen that when a; becomes — even, the infinitely remote terms must 



CO 00 



still diverge. 



In the Memoires on Series and Definite Integrals, which Poisson has published in different 

 Cahiers of the Journal de V Ecole Polytechniqiie, a fault analogous to that above noticed is very 

 frequently committed-f. It is the common practice of this distinguislied analyst arbitrarily to 

 introduce the ascending powers of a foreign variable, in connexion with the terms of an isolated and 

 independent series, and then to employ the extreme case of the general form thus obtained, when 1, 



or rather 1 is put for the new variable, instead of the original series. In this way he con- 



verts the neutral series 1-1 + 1 



1 H- &c. into a convergent series, and thus gets - for the sum : 



which is of course erroneous. He applies the same process to periodic series in general ; thus, in 

 fact, destroying their periodicity — at least in the infinitely remote terms — and tlience obtains sum- 

 mations that are palpably wrong. Thus, in referring to a particular series of this kind, in his last 

 great work, he says, " Elle est de I'espece des series periodiques, qui ne sont ni convergents ni 

 divergents, mais qu'on peut neanmoins employer en les considerant comme les limites de series 

 convergentes, c'est-a-dire en multipliant leurs termes par les puissances ascendantes d'une quantite 

 infiniment peu different de runite"! : the inaccuracy of which principle I have, I think, suf- 

 ficiently discussed elsewhere ||. 



It is of importance to observe, however, that thei'e is one class of series in reference to whicli 

 the adoption of this principle is allowable, as its application will be unattended with error : — I mean 



convergent series. For since, as already shown, the foreign multiplier 1 , becomes effective 



cc 



only in the terms infinitely remote, and as all these in converging series are themselves zero, these 

 multipliers produce no modification of the character of the series, nor any change in its sum. In 

 periodic series however error must of necessity arise from replacing them by the limits of converging 

 series; inasmuch as these latter always tend to some determinate value — either finite or infinite; 

 whereas an infinite periodic series, from its very nature, tends to indeterminateness. To attribute a 

 unique value to such a series is therefore absurd. 



I have here spoken of the sums of converging series as sometimes tending to injiuity, which 

 tendency some may suppose to be opposed to convei'gency : a simple reference however to the series 

 1 -h .r + x' + &c. will I think correct this supposition, since it will be admitted that this continues 



convergent for all values of .r from ,v = — up to x = \ : for which extreme value the sum is 



OS oo 



infinite 6. I have also ventured to call the infinites, to which the extreme cases of certain convergent 



• See also Proceedings of the Royal Irish Academy, 1846, 

 No. 49, where the communication referred to is printed at length. 

 + Journal de V Ecole Poly technique^ Cahiers 17, 18, and lli. 

 + Tliiorie de la Chaleur^ p. 199. 



Philosophical Magazine-, Dec. 184.'). 



i; The series 1 + - 



■J 1.2.3 

 for all real values of .r, and tends to infinity as x does. 



&c. also, is convergent 



