43 



extends beyond the series (11) to the right; a fact which is 

 of no moment when this term merges in zero, but of infinite 

 consequence when it merges in infinity. In such a case there- 

 fore, a numerical error, of infinite amount, is committed at 

 this step of the reasoning. Again, if the series within the 

 brackets at (13), have its terms, like those of the original, 

 tending to infinity, another numerical error of infinite amount 

 comes to be introduced; and so on. In fact, just as in the 

 method of definite integrals, before discussed, it is assumed, at 

 each step of the reasoning, that terms infinitely great are ex- 

 cluded ; and not only so, but that the terms ultimately dimi- 

 nish to zero. In the contrary case, therefore, the dilferential 

 theorem is altogether inapplicable, leading to results which 

 are equally inadmissible, whether the terms of the series in- 

 crease without limit, or remain stationary in value : forming 

 what has been called a neutral series. In this latter case the 

 error committed will be finite; in the former it will be infinite. 

 That an error is really committed in the application of this 

 theorem to neutral series, will be more explicitly shewn pre- 

 sently. 



Notwithstanding the imperfections noticed above, it should 

 create no surprise that, in the applications of the differential 

 theorem to particular diverging series, we so often obtain the 

 algebraic function whose development really gives rise to the 

 series, although no numerical approximation to the diverging 

 series itself. The function, whose development gives rise to 

 the series, being represented by /(a;), the series itself may be 

 represented by f(x) — ^{x), agreeably to what has already 

 been shewn in the former part of this Paper : it is the neglect 

 of the function (j>(x), in the particular application considered, 

 that introduces the infinite numerical error into (13) ; leading 

 us to conclude that, for the proposed value of a;, f{x) = s, in- 

 stead of/(x) — (p{x) = s. Now if there exist a convergent case 

 of s, that is a case in which (p{x) — 0, the differential theorem 

 will compute it, furnishing the proper function of x,f{x), 



E 2 



