Mr. DE morgan, ON DIVERGENT SERIES. 189 



2 /•" ,,, 2 /•« ., 2 /•« „ 



—z— I 6 '(bvdv + —y— I e-'d),vdv = — -— / 6"''"d« = 1. 



Accordingly, if all that precedes be correct, we have 



which is certainly false, unless a convergent series can represent less than half the sum of its terms. 

 This last series is always convergent, except only when a = 0, or < = 1, in which case the last 

 equation is found to be algebraically true. If for a; we write - x, the equation is found to be 

 true when t is equal to the least of x and cv'\ but is certainly not universally true. 

 Apply the same process to 



(1 — x) cosav 



1 - 2,r cos2rau + a?" 

 and the result is 



cosa w + x cos Saw + x- cosSau + 



'-h'^'^[''^i^}'H'''^i^''' 



on which precisely the same remarks might be made. I might multiply instances of this kind to 

 any extent ; but the following consideration will render them needless, as showing that what we 

 have seen is precisely what we ought to have expected. 



Integration, though only capable of an arithmetical definition, is the most decided changer of 

 form which we ever use. A change of value in a constant may introduce a totally different form 

 into an integral ; and in particular, the assumption of infinite value for a constant has this effect 

 almost without exception. And in regard to definite integrals, there is hardly any end to the 

 known instances in which complete and apparently arbitrary changes of form (such as cannot pass 



one into another through - or the like) arise from alteration of the specific value of a constant. 



If then V be expanded into the series /•„ + ^i + A + ••• and if the sum of n terms, 

 ^0 + ^1 + ... + Pn- i ''e called Q„ ; we obviously have 



jydv^^P.dv + I'p.dv + ... + L\V-Q„)dv 

 where w is made infinite after integration. When the series Pq + P, + ... is convergent, then, even 

 granting that y{V — Q^) dv may have circumstances peculiar to w = oo , it is of no consequence, 

 since considerations of form are rendered useless by evanescence of value : the elements ofy( V- Q„) dv 

 must, by the hypothesis of convergency, diminish without limit as compared with the corresponding 

 elements oifP^dv, fP^dv, &c. Even if integration converted the convergent series into a diver- 

 gent one, this would still be the case. But if P„ + P, + ... be divergent, we have no longer any 

 right to draw any conclusion about /"( F — Q,) dt) from observing what takes place -vihh J'P^dw, 

 J'P\ dv, &c. Applying this to our first example above, we have 



1— <rcosa« cos (n + 1) av — X cos nav 



= 1 + a? cos av +... + a!"coswa« + of*' 



\ -2xcosav + x' I -Zxcosav + X 



change .i' into a;"', and add ; which gives 



1=2 + 



+ '+ lcos(w+ \)av- Ix"*- + — )( 



lx + -] cosav + . . . + ix"+ — I cos no v h 



1 — 2.1? cos au + X 



This equation is identically true, the only restriction being that w must be a positive integer 

 (0 included). Consequently, wt liave, as specimens of legitimate inference from integrating a 

 divergent series, 



BB 2 



