268 THIRD REPOIIT — 183o. 



being 2>itid in such cases to terms which are at an infinite di- 

 stance from the origin. 



It is this last condition, which, though quite indispensahle, 

 is rathei' calculated to ofiend our jiopular notions of the values 

 of series as exhibited in their sums. We speak of series as 

 having sums when the arithmetical values of their terms are 

 considered, and when the actual expression for the sum of n 

 terms does not become infinite when n is infinite, or when, in 

 the absence of such an explicit expi'ession, we can show from 

 other considerations that its value is finite. In all other cases 

 the series, arithmetically speaking, may be considered as di- 

 vergent, and therefore as having no sum *, if the word sum be 

 used in an arithmetical sense only, as distinguished from gene- 

 rating function. 



We are in the habit of considering quantities which are in- 

 finiteljj great and infinitely little as very differently circum- 

 stanced with respect to their relation to finite magnitude. We 

 at once identify the latter with zero, of which we are accus- 

 tomed to speak as if it had a real existence ; but if we subject 

 our ideas of zero and infinity to a more accurate analysis, we 

 shall find that it is equally impossible for us to conceive either 

 one or the other as a real state of existence to which a mag- 

 nitude can attain or through which it can pass. But it is the 

 relation which magnitudes in their finite and conceivable state 

 still bear to other magnitudes in their course of continued in- 

 crease or continued diminution, which enables us to consider 

 their symbolical relations when they cease to be finite ; and 

 whilst quantities infinitely little are neglected as heing absorbed 

 in a finite magnitude, so likewise finite magnitudes are consi- 

 dered as being absorbed in infinity, and therefore neglected 

 when considered with relation to it. The principle, therefore, 

 of neglecting terms beyond a finite distance from the origin, in 

 converging series, is both safe and intelligible, whilst the case is 

 very different with respect to the neglect of similar terms in a 

 diverging series. Of such series it is said that they have no 

 arithmetical sum ; but it may be said in the same sense of all 

 algebraical series involving general symbols that they have no 

 sum. But it is not the business of symbolical algebra to deal 

 with arithmetical values, but with symbolical results only ; and 

 svich series must be considered with reference to the functions 

 which generate them, and the law^s of the opei-ations employed 

 for that purpose. The neglect, therefore, of terms beyond a 



* This would appear Cauchy's view of the subject : see the 6th chapter 

 of his Cours d' Analyse, 



