Mr. DE morgan, ON DIVERGENT SERIES. 197 



divergency, has a tendency to destroy the exclusive confidence which many modern analysts have 

 placed in them. 



The very foundation of this confidence is, as we have seen from the expressions of Poisson, 

 a full belief in the maxim that whatever is numerically true up to the limit is true at the limit. 

 To this principle, reasonable and convincing as it is, let us join the remembrance of a fact so 

 well ascertained, were it merely as a matter of observation, that alternating series are more safe 

 and more easily calculated than progressing series, and also the simplest of all theorems on 

 convergency, namely, that an alternating series is rendered convergent by mere diminution, if 



unlimited, of its terms. With these premises let us consider the integral / '- d.v. I 



believe that this one integral might be made to throw a case of exception in the way of those 

 who have claimed privileges for the finitely diverging series over other non-arithmetical forms 

 in every particular as to which their superiority has been asserted. 



Poisson, agreeing in this point with all other analysts, asserts that / d.v is J^ tt, o 



or — ^TT, according as a is positive, nothing, or negative : any computer using the method of 

 quadratures would confirm this result in all its parts. But this integral is clearly the same as 



/""sinaa; ["sina.v fosina.v 

 Jo dx + J„ d.v + J2n d.v + . 



which is an alternating series, since the second, fourth, &c. integrals are composed entirely 

 of negative elements. Moreover the terms diminish without limit, since the numerators of the 

 elements are recurrent, but the denominators constantly increasing, and without limit. However 

 small a may be, if it be positive, ^ tt is the real value of the series, obtainable by the computer : 

 and yet if a be absolutely = 0, each of the terms is also absolutely = 0. But if 1 — I + 1 — ... i.s 

 to be taken as having the unique value ^, lohich may be employed in analysis (the Italics are 

 Poisson's expressions) because 1 - g + g' - ... is certainly (1 +§■)"', however little g- may fall short 



of unity, then surely 0-0 + 0-0 +... may here represent either — - or + — , since, however 



small a may be, when negative it gives the first, and when positive the second: notwithstanding 

 which, it is certain that - + — ... is in this case = 0. 



Here then we have — + — ..., a limiting form, and that which is true up to the limit is 

 not true at the limit. But why is this principle abandoned, being, as it is, the very point on the 

 assumed clearness of which the line is drawn between the accepted and the rejected cases of non- 

 convergeney ? Is it that an infinite series of zeros must represent zero? I think I have shown 

 .sufficient cause against that assumption. Is it by the principle of mean value discussed in the 

 last section .'' No one that I know of, except Leibnitz on grounds purely metaphysical, has ever 

 used this principle, and no one has hitherto stated it in general terms : and moreover the modern 

 analysts appear to require strictly arithmetical foundations, and would acknowledge no identity 

 of principle between their methods and one which produces tan oo = v — 1 i they seem also to 

 suppose that they are quite free of the use of principles established by induction. Either then 

 the principle that whatever is numerically true up to the limit is to be held true at the limit 

 must be abandoned, or exceptions of discontinuity, in questions involving integration, must be 

 admitted to be possible in a manner which renders the cases to which Poisson and others have 

 confined themselves subject to as great difficulties as those which they have abandoned. 



In a preceding part of this paper I spoke of it as a strong ])resumption that A„ + Ayr + A.,.v- + ... 

 xhould represent A^^ when .v - 0, or that the form + + + ... whicli follows A^^ should =0. 

 If A^^, A,, &c. be all positive, and if the series be always divergent, however small ,v may be, 

 It it obvious that whore the preceding represents a function of complete continuity, we may 



cc 2 



