Mr. DE morgan, ON DIVERGENT SERIES. 183 



We must admit that many series are such as we cannot at present safely use, except as means 

 of discovery, the results of which are to be subsequently verified : and the most determined rejector 

 of all divergent series doubtless makes this use of them in his closet. But to say that what we 

 cannot use no others ever can, to refuse that faith in the future prospects of algebra which has 

 already realised so brilliant a harvest, and to train the future promoter of analysis in a notion 

 which will necessarily prevent him from turning his steps to quarters from whence his predecessors 

 have never returned empty-handed, seems to me a departure from all rules of prudence. The 

 motto which I should adopt against a course which seems to me calculated to stop the progress of 

 discovery would be contained in a word and a symbol — remember ■^y'—l. 



I do not pretend to have that confidence in series which, to judge from elementary writers on 

 algebra, is common among mathematicians : not even in convergent series. A few great forms, 

 which have had substantive and finite expressions assigned to represent the remnants after any 

 given term may, no doubt, be perfectly trustworthy. But as for the rest, I cannot bring myself 

 to that positive assurance with respect to any general class of series which the writers to whom I 

 shall presently allude appear to have with respect to such divergency as they do admit. The 

 main object of this paper is to show that they have underrated the character of most of what 

 they reject, and overrated that of all they receive. 



I shall now proceed to the different points of discussion in order. 



SECTION I. 

 Alt Divergent Series, whether their divergence lie finite or infinite, stand vpon the same 

 basis, and ought to be accepted or rejected together, as Jar as any grounds of con- 

 fidence are concerned which are not directly derived from exjieriefice. 

 I SHALL first examine the general arguments on which Poisson supports the contradictory of 

 the preceding assertion. This great analyst was at the head of the school in which definite integration 

 had been made in a great measure to take the place of expansion into algebraical series. A definite 

 integral is a particular kind of series, and has its converging and diverging cases, the latter being 

 either of infinite or of finite divergence. Thus j^ 6 dx is convergent, _/„ e' dx is infinitely 

 divergent, and y,, cos.rd^i? is finitely divergent. Perhaps in the natural bias derived from a 

 continual contemplation of integration under the form of summation, not of inverse differentiation, 

 may be seen the reason for the opinion of divergent series adopted by the definite integrators. 

 Let it only be granted that integration is as fully defined and as generally understood, as any of 

 the fundamental operations of arithmetic, and the question on diverging series seems to be settled 

 at once, and by a much easier argument than any of those usually proposed against them. To 

 take an instance; — j„ 2'dx cannot be other than /„ a'da; + j, Z^dcV + J^ S'dx + ... : but the 

 first is (on the above assumption) infinite, and the second is (log 2) ~'(l + 2 + 4 + ...) which is 

 therefore infinite. Consequently I + 2 + 4 ... cannot, as usually held in algebra, represent - 1. It 

 must certainly be charged upon those who have hitherto used divergent series, that they have 

 never reflected upon and explained, perhaps have never perceived, the singular apparent in- 

 con-sistency which they were every day committing; namely, treating those very forms as repre- 

 sentatives of infinity when they were consequences of integration, which they accepted as finite, 

 when they were results of algebraical development. Referring further discussion of this point 

 to a subse(|ucnt section, I now make two citations from memoirs by Poisson in the Joiirnn/ 

 de PEciile Polytef/mii/ue, Cahier 1<), pp. 408, 40.9, 501. 



Page .501 " On enseigne dans le.s elemcns, qu'une serie divergcnte ne pcut servir il calculer 

 la valeur approchee de la fonction dont elle resulte par le developpcment : mais quelqiiefois on 



