196 Mr. DE morgan, ON DIVERGENT SERIES. 



J„ A-i &c. increase, the more rapidly does this diminution take place. We shall see, in the next 

 section, that this comminuence oi Ar, - AyX + ... and .r"' is to be looked for as the rule, its failure 

 being the exception. 



Let the series be transformed into 



1 f X x^ 



\A„ - (A, - aA„) + {A2 - 2a A, + a'A) ,— tj 



1 +aw\ ' (I +ax) (1 +aa!y 



(J3 - 3a ^2 + Sa'Ai - a'Ao) 



...( 



(1 + a.vy 



which is easily done. Let a be taken so small that the series just obtained shall still be alternating, 

 which can generally be done, though not always, and then, on account of the factor (l + a,r)~\ it 

 is clear that the original series and *'"' are comminuent except only when the second series and .r 

 increase without limit together : that is, instead of supposing, as a priori we should do, that the 

 alternating form with terms increasing without limit has an equal facility of approaching any given 

 limit, we are rather to look upon it that its facility of approaching any other limit except 0, as ,r 

 increases without limit, is only equal to that of its approaching os , or increasing without limit. I 

 am not, of course, disposed to attach much weight to reasoning which rather resembles that of the 

 theory of probabilities than of pure mathematics: but I do say that it must be better to take such 

 considerations at their proper value, as suggestions for the conversion of results of observation into 

 demonstrated theorems, than to allow isolated facts which evidently point at something, to remain in 

 their state of separation. 



This inaptitude to represent infinity, and this tendency to comminuence with a-"' are both cir- 

 cumstances which render the operation of integration much safer as applied to alternating than to 

 progressing series. But the principal distinction between the two kinds of series seems to me to 

 depend upon our present knowledge of the meaning of integration, as explained in a previous 

 section, being imperfect. The progressing series cannot be expressed differentially without the 

 operation of what we may call progressing integration ; the alternating series can. This is exem- 

 plified in the two following remarkable theorems, given by Poisson : 



^0 + (p\ + (p2 + ... =^<pO + f'(t>zdz + 22^;" ^fo" cos 2mTr z (pzdz}, 



(po - (pi + <p2 - ... = 100 -1- 22™if {7^" (cosmnz - cosSmTr;^) <pxdx\. 



We may now examine the sort of proof which we can obtain of the usual values of divergent 

 series, with the view of comparing finite and infinite divergence. Let F = /"„ - P, + P^ - ... and 

 let P„ = 1 when .i- = 1, independently of re. Also, before *■ = 1, let the series be convergent ; after- 

 wards divergent. Let P„ = P„_, - p„_,, whence p„_, = when a; = 1. And 



F = P„ - (P„ - p,) + (P, - p,) - ... or F = 1P„ + (jD„ -p^+p,- ...). 



Again, let W = Q„ + Q, + Q^ + ... and let Q„ = 2" when ,r = 1. Let Q„ = 2Q„_] - 9„_, ; 

 whence q„= when x = \. And IF= Q„ + (2Q„ - q^) -(- (2Q, - 9,) -t- ... or W = - Q^- (<y„ + q^ 

 + q.;, + ...). When x = 1, we have 



F = 1P„ + (0-0-fO- ...) ; W^=_i_ (0-1-0 + 0+ ...) 



and on the proper equivalents of the two evanescent forms it depends whether 1 - 1 + ] - ... = -^ 

 and I +2+4 + ... = -1 are true or not. Now instances enough may be produced in which 

 + + + ... is not an equivalent of : though, by instances merely, it would be found exceed- 

 ingly difficult to overturn - + ... = 0, as long as the common operations of algebra only are 

 used. But here again, when the forms of the integral calculus are employed, instances may be 

 produced in which, though the form - + ... may still be called 0, it is only by means of a 

 discontinuity which, occurring as it does at the limiting form of an alternating series of finite 



