18ti 



Mr. DE morgan, ON DIVERGENT SERIES. 



until as general an invelopnient is found ; after which, I incline to think that all conclusions upon 

 the series should be upon them considered as the developments of those particular functions which 

 produce them. My reasons are as follows. 



1. Discontinuity of form is not perceptible in the series itself, though it may very possibly 

 exist ; to reason upon a series as a continuous function, without knowing from its invelopment that 

 it is so, is pure assumption. This remark applies particularly to series which are always conver- 

 gent, and most of all to series which are convergent and also begin to diminish from the first term. 

 If we spoke of mathematical results in the same sort of language as of physical phenomena, we 

 should say that there is inaptitude in developments to be the permanent arithmetical representatives 

 of finite continuous functions, and that series which must of necessity be always convergent, shew 

 this inaptitude by discontinuity, while the others escape from arithmetic altogether by divergency. 



■2. When divergent series are employed independently of their invelopments, it is impossible to 

 distinguish the cases in which they really represent infinity from those in which they are developed 

 forms of finite quantity. No one can actually calculate with the symbol x , even when its sign is 

 determinate : for even if eo + eo ' and as x co ' would not puzzle him, it is certain that 05 - co ' 

 and eo -^ eo ' would require reference to the producing functions. As soon as x is attained, we 

 must stop for examination : this cannot be done if, when attained, it is seen under the divergent 

 form which equally belongs to finite quantities, that is, is not seen at all. 



3. It cannot be questioned that series which are infinitely divergent, at least, may appear 

 as very different things in different cases. For instance, an algebraist would be inclined almost to 

 assert that 1 + 2 + 4 + ... must be - 1 ; for he would say, if it be the object of algebra at all, it 

 must satisfy the equation z = 1 +Qz. But now let us consider the series 1 + 2a-" + 2'a'"' + 2" a'"' 

 + ... which is certainly convergent, if n and n be both greater than unity, and as certainly increases 

 without limit, as c - 1 diminishes without limit. When « = 1, the limiting form 1 +2 + 4+ ... is 

 clearly the representation, not of - 1, but of 05 . The series e''' + xe''"' + x-e'''"' + ... satisfies the 

 equation 



ndU r" 



= 7 7^ or C7= / (he.x' 

 an -J. 



fi*S 



'd9. 



dxdb h dn 



where <ht) and \S/Q are arbitrary, and a and /3 are any constants independent of a, b, and n. In 

 taking this form for U, I follow the example of Poisson, Cauchy, &c., who are always content with 

 such a form, provided only that it contain the requisite* number of arbitrary functions. To make 

 the form of U an algebraical equivalent of the series, we must determine cpQ and \|/0 from 







b1''d9; 



a useless attempt, even when w<l, unless discontinuous forms of \^0 be introduced. Here is a 

 clear case in which 1 + 2 + 4 + ... represents eo : are we then really to abandon the assertion that it 

 satisfies the equation 1 +2% = x? If so, the opponents of divergent series have gained their point, 

 for those developments are not even to be trusted as to their symbolical properties. But I rather 

 argue that it is not so, in the following manner. Every equation, it is very well known, has as 

 many roots as units of dimension, only on the supposition that its problem is absolutely of 

 that dimension, and not a degenerate case of a higher dimension. Plenty of simple problems may be 

 proposed which illustrate this known result of common algebraical reasoning. Now the equation 

 which stands related to the series in question in the same manner asl+2« = «tol+2 + 4+ ... is 

 <pz = tv'e'''"' + (pix + I). If this last could be generally solved, then ^0 would be the series 



• They assume that Sf/>e£»', or <^, 6, c-**' +(^26,6^9^+ ... can 

 alway.s be represented by/f/iW (-''^r/P, which I believe to be true if 

 (/)C may be discontinuous. But it has not been proved ; should it 



happen to be false, * toujours incertains' may be applied to many 

 of their results, and 'le plus souvent inexncls,' may follow. 



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