Mr. DE morgan, ON DIVERGENT SERIES. 195 



sequently \|/« + v|/ (u + 1) must be always positive from « = .r to y = os . Hence v|/j) cannot 

 change from + to — when v = a, without changing again from — to + before v = a + 1. Now 

 the theorem can only be disturbed by \f/ ?i becoming negative: for xl^.r = Q!),r - \|/ (*• + 1), or, 

 xj/ (a; + 1) being positive, \^x >(px ; again \\,x = <pa; - <p {w + \) + y^ {w + 2), or, \\/ {x +2) being 

 positive, \//.r; < (pw — (p{x + 1), and so on. 



Hence 1. No function y^x can be expanded as above unless it be one in which its changes of 

 sign go in pairs, the — I- change following the h — change before the variable has received an 

 additional unit : 2. except at those epochs at which \^ (x + n) happens to be neg-ative, the theorem 

 must be true. As long as <px, (p {x + 1), &c. continue diminishing, the theorem must either be 

 true, or there must be a minimum value of ipx within a unit-change of the variable, reckoning 

 from the last change of sign. When \^x changes from + to — , \p'x is negative, and when from 

 - to +, y^'x is positive: there must then be a minimum value of \px between the two changes. 

 Now as long as (px diminishes, or cp'x is negative, \\/' x + y^' {x + I) is also negative. After the 

 minimum is past, then, yp'x cannot continue positive until x has increased by a whole unit, or there 

 must be a maximum value within a unit-change of the variable, reckoning from the minimum. If 

 then the terms continue diminishing as far as (p (x + n), it may be collected from the above that the 

 theorem is true for the several summations up to (p(x + n - 1), except for those in the neighbourhood 

 of the last terms of which are found two roots o{\px for values of x not differing by a unit, followed 

 by a maximum value of ypx, for a value of x not a unit in advance of that which gives the inter- 

 mediate minimum of the roots. And if \jy x can ever become infinite, (px being finite, then ... 

 ■^{x + 2), yfr {x + 1), yj/x, \p {x - 1), ... are all infinite, with alternate signs. From this it will readily 

 be seen that in the greater number of cases the theorem must be strictly true. 



Again, it is now known that every function (px can be expressed in the form '2Je"', provided that 



integration be included under the sign 2, and also the finite summation of terms in which A is 



infinitely great, and a infinitely small, and which give a finite sum by difference of sign. 



Whether many cases of this reduction do not involve much greater difficulty than those of divergent 



series, may be a question. However this may be, it is clear that in whatsoever manner cpx may 



be represented by 'S.Ae"', in the same manner (px - (p (x + 1) + ... may be represented by 



Ae" 

 2 ^. In all cases, then, in which the several terms of 'EAe"' are severally positive, and, if 



Ae"'' 



infinite in number, can be arithmetically summed, it follows that ypx or E ^ is also positive. 



1 + e" 



Thus for all cases in which (px can be expressed by J^e"''y_vdv, -^v being always positive between 



the limits, it follows that the theorem is true. 



We find then that this theorem must be true in the great majority of cases: as far as 

 observation goes it is not known to have failed in any one of the instances in which its use is of 

 importance. It is enough, without any thing else, to draw a great distinction between the pro- 

 gressing and alternating series. But this is not all : it is also matter of observation that there is 

 great difficulty in finding alternating series which become infinite for one or more values of their 

 variable, without having recourse to those in which the law of the coefficients is discontinuous. It 

 is most easy, both to make the above theorem fail, and to procure a case in which infinity of value 

 can be obtained, by means of the development of common algebraic functions, presenting discon- 

 tinuous coefficients; but it is not easy with coeflScients following a continuous law. 



It cannot of course be proved that A^- Af + A^~ ... is necessarily a finite quantity, since cases of 

 exception may be procured : but some illustrations may be given of the tendency of this form to 

 represent only finite quantity. I'robably notiiing l)ut the collection of such fe?tdenrien will ever 

 lead to a rigorous criterion for ascertaining in what cases it can represent infinite quanlity. 



In a great many cases, a large majority of those usually considered, the complete alternating 

 Her'ien A„- A,x + AjX' - ... diminishes without limit, as ,» increases without liinit : and tiie faster 



Vol.. VIII. I'Aiir II. Cc 



