188 Mb. DE morgan, ON DIVERGENT SERIES. 



as solved, and let the process of solution have a name. But the state of our knowledge makes it of 

 no use whatever to express a conventional solution, since our power of translating our convention 

 into ordinary language is confined to a small number of cases, all rendered backwards from the 

 direct process. Common integration is only the memory of differentiation : and the process of 

 parts, and the few other artifices by which it is effected, are changes, not from the unknown to the 

 known, but from the forms in which memory will not serve us to those in which it will. We may 

 assume that any function has an integral, and we may write down fcoiw'dx or fe''"da!\ we 

 may also have recourse to series, and by assuming an unlimited use of divergency, we may procure 

 abundance of nominal answers to any question. But we cannot be so much as sure of the fact that 

 every continuous function has an integral, except by recourse to the summatory definition, namely, 



/•' f / w - a\ I a; - a\ , / x - a\\ a; - a 



j^ct>vdv = \^<pa + <p[a+-^^^ +cp[.v + ^—-^ +...+ ,^ U- + „ __ j J __ 



in which n is made infinite. This definition, as is well known, never fails, nor can fail, to give one 

 value for every value of a and .r, applied to one branch of the function, except only when (px^ 

 becomes infinite at or between v = a and v = x. In this last case, we have not even the means of 

 universally defining fcpvdv : all the difficulties of divergent series meet us again. 



In confining ourselves to this arithmetical definition of an integral, when one of the limits is 

 infinite, we must, as to a large number of cases, act precisely as if we separated a class of divergent 

 series from the rest, and insisted upon their retaining for their values the idea which the attempt at 

 arithmetical summation gives, infinity. The early problems by which the nature and use of 

 integration is suggested, being problems on concrete (mostly on space) magnitude, cannot afford the 

 means of generalizing our definition. No doubt the area of the curve y = €^ represented by 

 f" edx, is greater than any surface which can be assigned : no doubt also that the series of inscribed 

 rectangles 1 + e + e^ + ... is the same. AVhen we shall have obtained the definition of an integral 

 by which we can state such a value for j'^e'dw as is the true correlative to (l-e)"' considered as 

 the value of 1 + e + ... then, and not till then, shall we be entitled to claim integration as an 

 instrument of algebra in the widest sense. Some of the objections raised against divergent series, 

 indeed most of those which are very plausible, are grounded upon the supposition that integration 

 may be as unreservedly applied to divergent as to convergent series, if the former are to be used 

 at all. That this cannot be done may be satisfactorily shown by instances, as follows : 



\ — X cos av , 



Let (bv = = 1 + acos av + x' cosSati + 



' 1 — 2 a; cos av + x' 



which never becomes infinite for any value of v, except only when a; = ± 1 ; and the series is con- 

 vergent when X lies between — I and + 1. Multiply both sides by e'^^dv, and integrate from 

 ti = to «= 00 , in which case there cannot be any doubt about the purely arithmetical (or convergent) 



character of every integration. This gives us, t being 6 ■• 



2 



-7- f €-'''(pvdv = i +xt + a^ t^ + a^ f + x^t^' 



This resulting series is convergent for all values of x : for, since t is less than J, xt" must 

 become less than unity after a certain value of n, and thenceforward Sixt")" must be more con- 

 vergent than any series of simple powers. If x lie between - 1 and + I , the whole of this process 

 is purely arithmetical, and the identity of the two sides of the last equation might be approximately 

 verified by actual computation : if not, the original series, though divergent, is changed into 

 a convergent one by the process. Change x into x~\ and let (pv then become (f>]V ; we find 

 (bv + <biV = 1, and 



