Section 1. — All divergent series, whether their divergence he finite 

 or infinite, stand upon the same basis, and ought to be accepted or 

 rejected together, as far as any grounds of confidence are concerned 

 which are not directly derived from experience. The author exa- 

 mines the reasons which Poisson gives for maintaining the contra- 

 dictory of the preceding. That great analyst considers 1-1 + 1 

 — 1 -t- ... for instance, as 1 —ff + g- — g 3 + . . . , where g is less than 

 unity by an infinitely small quantity. Mr. De Morgan maintains, 

 that this method, if allowed in transformation of a finite diverging 

 series into a convergent one, of which the convergency only begins 

 after an infinite number of terms, must also be allowed, unless rea- 

 son can be shown against it, in the destruction of the infinite cha- 

 racter of an infinitely diverging series, by the tacit retention of the 

 infinite remainder after an infinite number of terms. 



The author would not use any series, so as to place absolute de- 

 pendence upon their results, unless the producing functions were 

 known : and this because series themselves neither show disconti- 

 nuity nor infinity, when it takes place ; and because it happens that 

 divergent series, at least, and perhaps others, may represent one thing 

 or another, according to the general form of which they are made 

 particular cases. 



Mr. De Morgan observes that a divergent series, which is not 

 considered as arithmetically infinite, such as 1+2 + 4-)- . . . maybe 

 so in reality, in particular cases. This series being called S, satisfies 

 the equation S =1 + 2 S, and this gives S = — 1, the usual value of 

 the series. But it is to be remembered that an equation may be a 

 degenerate case of an equation of higher degree, in which case it has 

 one or more roots infinite. An instance is produced in which 1 + 2 

 + 4+ . . . certainly represents infinity. 



Finally, the author remarks that there is much more safety in se- 

 ries with terms alternately positive and negative, whether their diver- 

 gence be finite or infinite, than in series of finite divergence, as such. 



Section 2. — The operation of integration, as at present understood, 

 is one of arithmetic, as distinguished from algebra, and must not be 

 applied unreservedly to divergent series. The author supports the 

 first part of this assertion upon the circumstance that the only defi- 

 nition of integration which is generally applicable is the summatory 



one, in which j <f> x dx does not mean the function whose differen- 

 tial coefficient is $ x, but the limit of the summation expressed by 

 S (<px Ax). He then goes on to show instances in which it is un- 

 questionably not allowable to apply integration to infinitely diver- 

 gent series : and he asserts throughout the paper generally, that all 

 the instances in which error has been shown to arise from the use of 

 infinitely divergent series, have been those in which integration has 

 been applied, and those only. 



In this section warning is also given against the supposition that 

 + + 0+ . . . must represent in all cases. 



Section 3. — It generally happens that the real analytical equiva- 

 lent of the different values of an indeterminate expression, is the mean 



