Mr. DE morgan, ON DIVERGENT SERIES. 187 



required : if, after solution, b were made = 0, we should see that 1 + 2ss = z, the result for .v = 2, 

 would be only a degenerate form of a more complicated form. 



This remark will illustrate my opinion that a series is to be considered strictly in relation to the 

 function from which it is developed. If x' + af*^ + ... be absolutely under consideration, the 

 equation (b« = ar(l - x)'' may be strictly obtained, and thence (l - a;)"' for 1 + ,r -t ... But 

 there is no saying what further degeneracy of form may be seen in passing from cbz = x'e~'"'' + 

 <p(x + 1) to 1 + 2^ = 21, which is not seen in passing from ^2; = x' + (j){z + 1) to the same. 



My conclusion is, that a divergent series may have for its proper value either that which is 

 usually so considered, or infinity, according to the nature of the function from which it is expanded. 

 And since every equation has as many roots as it has algebraical dimensions, so many of them being- 

 infinite as there are vanishing coefficients which precede the first finite coefficient, there can be 

 no right to say that the symbolical character of divergent series is forfeited, until either the symbol 

 w takes the place of the ordinary value in a case in which there is no degeneracy, or until some 

 finite value, different from the ordinary one, is shown, in some one particular case, to be the proper 

 representative of the series. Let 1 +2 + 4+ ... be shown to be any thing but a root of either 

 \ +2z = ss, ov of another equation which has degenerated into 1 + 2« = « ; that is, let it come out 

 any thing but - 1 or w , and as a result of any process which does not involve integration performed 

 on a divergent series — and I shall then be obliged to confess that divergent series must be aban- 

 doned, or rather, that the generalizations frequently made on the subject must be much curtailed. 

 But nevertheless, there is nothing to lead us to doubt that divergent'series of all classes, whether of 

 finite or infinite divergence, must be treated alike. If any one say that such a difficulty as the 

 preceding cannot occur in series of finite divergence, he must prove it. 



It might perhaps be supposed that, in every doubt which has been raised in the preceding 

 remarks, the finitely diverging series have been much less hardly borne upon than the others — to an 

 extent which may make it seem to be almost admitted by myself that the foreign analysts, if not 

 justified in their dogmatical rejection of infinitely diverging series, have nevertheless chosen, and 

 judiciously chosen, to confine themselves to the safer of two paths. But it is to be remembered 

 that I have been obliged, as yet, to mention only their practical division, which really consists in 

 the separation of all finitely diverging series from the rest. Had I had to make my own division 

 of series, I should have admitted that there was one of two paths which was much safer than the 

 other : but I should have asserted that the labors of the writers in question did not extend over the 

 whole of that path. From the sort of appeal to induction which unfortunately must, in the present 

 state of our knowledge, help us to a part of our results on series, backed by considerably more of 

 demonstration than has been applied to the remaining cases, it seems to me pretty clear that the 

 proper line of demarcation does not separate series of finite and infinite divergence, but series iiaving 

 all their signs alike from those of terms alternately positive and negative, or consisting of parcels 01 

 terms alternately positive and negative. This will be the subject of a subsequent section. 



SECTION II. 



Tlie Operation of Integration as at present imderstood, is one of Arithmetic, as distin- 

 fruisliedjrom Algebra, and must not he applied unreservedly to Divergent Series. 



AccoBDiNc to elementary notions, we differentiate when we find the value of \ip(!v + h) - (p.v\ h~ ^ 



in a calculable form when A = 0. Integration is usually defined as the inverse question, which 



must be, re(|uired (^.i when the calculable form of \(f) (x + h) - <^.r| A"' is given for li = 0. This 



demands the solution of a functional equation, and it is easy to say. Let this ecjuation be considered 



Vol.. VIII. I'Aicr II. Bb 



