XV. On Divergent Series, and various Points of Analysis connected with them. Ey 

 Augustus De Morgan, V.P.R.A.S., F.C.P.S., of Trinity College; Professor 

 of Mathematics in University College, Londoji. 



[Read, March 4, 1844.] 



I BELIEVE it will be generally admitted that the heading of this paper describes the only subject 

 yet remaining, of an elementary character, on which a serious schism exists among mathematicians 

 as to absolute correctness or incorrectness of results. When such a question arises upon a method 

 of pure mathematics, there can be little doubt that it must be one which is likely to lead to error 

 if not cautiously used ; and it is probable that the contending parties have not made any close 

 agreement upon the use of terms. A review of the leading points of the controversy may be useful, 

 accompanied by an examination of the maxims which have been adopted, but I think not very 

 plainly stated, in the rejection of the series called divergent. The manner in which the rejection 

 just alluded to has been made will require that, instead of dividing series into convergent and 

 divergent, we should make a more general division, say into convergent and non-convergent. 

 Non-convergent series may be divided into those of infinite and finite divergence : the former of 

 which, as in the cases of a + a + a + ... and 1-2 + 3-4. + .. . can be made, by summation of 

 terms, to differ from a given quantity to any extent; the latter, as in the cases 1 - 1 + 1 - ... 

 and COS0 + cos2 + ... cannot be made to differ from a finite quantity by more than an amount 

 which can be ascertained. It is obvious that only the converging series can, properly speaking, be 

 the objects of arithmetical calculation, in which they occur early, of which ^ = '33333... is a 

 sufficient instance. All others, whether of finite or infinite divergence, are equally out of the pale 

 of arithmetic to those who do not acknowledge different degrees of impossibility. I do not here 

 argue with those who reject everything which is not within the province of arithmetic, but only 

 with those others who abandon the use of infinitely diverging series, and yet appear to employ 

 finitely diverging series with confidence. Such appears to be the practice of those analysts 

 who object to diverging series, both at home and abroad. They seem perfectly reconciled to 

 1-1 + 1-1 + .. .=1, but cannot admit 1+2 + 4 + 8 + ... = -1. 



Many of an earlier school took an opposite position ; they freely used infinitely diverging 

 series, but, with Euler, considered finitely diverging forms as indeterminate. To use a common 

 phrase, they spoke as they found : they could actually obtain by rules of algebra, finite expres- 

 sions from which they could evolve infinitely diverging series : but they were not able to find, or 

 to satisfy themselves they had found, similar equivalents for most cases, particularly the trigono- 

 metrical ones, of the remaining species. They made an unguarded use of the word ' indeterminate :' 





 sometimes it meant nndeterminable, in the same manner as - when looked at as the solution of an 



identical equation ; sometimes only undetermined, either with reference to the state of science at 

 the time the word was used, or to the state of a particular question at some one particular stage of 

 the solution (as in the method called that of indeterminate coefficients). The moderns seem to me 

 to have made a similar confusion in regard to their rejection of divergent series : meaning sometimes 

 that they cannot be safely used under existing ideas as to their meaning and origin, sometimes 

 that the mere idea of any one applying them at all, under any circumstances, is an absurdity. 



