IN REFERENCE TO CERTAIN RESULTS OF ANALYSIS. 433 



scries thus tend, determinate : because if we reflect upon the peculiar character of a strictly diverg- 

 ent infinite series, we shall perceive, that however remotely into the region of infinity its terms be 

 considered to extend, yet we can never, even in imagination, reach a stage beyond which the series 

 ceases to be accumulative, and may be rejected as zero: the portion so rejected would, on the 

 contrary, still be infinite ; and this is a peculiarity which sufficiently distinguishes a divergent 

 series from a convergent series with an infinite sura. It lias place even in those slowly diverging 

 series of which the individual terms continually tend to zero, as, for example, in the series 



' 1 1 1 „ 



1+- + - + -+- + &c. 

 2 3 4 5 



12 3 4 



and h + h h &c. 



1.3 3.5 5.7 7.9 



for however remote the «"' term may be, n terms more of the first of these series will be 



1 1 



n + S 2n 



and M terms more of the second, 



n + I w + 2 2n 



+ 



4(n + 1)^-1 4(w + 2)^-1 4(3n) 



and these additional w terms will, in the first case, exceed 



1 1 



w X — 

 2n 



and in the second case. 



1 



in 



A diverging infinite series therefore tends to no limit, either finite or infinite ; and this 

 consideration is perhaps sufficient to justify the language of the continental analysts, who say that 

 such series have no sum. 



It would seem desirable however to divide series into other classes besides convergent, divergent, 

 and periodic ; in order to distinguish those which come under the influence of continuity, from 

 those which, like the series just considered, are entirely isolated and independent. The latter class 

 might be called independent or neutral series; and the former dependent series. Hutton* appears 

 to have called the series 1 - 1 + 1 - 1 + 1 - &c. a neutral series, simply because it is neither 

 convergent nor divergent. In the sense in which it is here proposed to use the term, no reference 

 is made either to convergency or divergency : but merely to the fact of the series not being united 

 to a set of others by the bond of continuity. A neutral series may therefore be either convergent, 

 divergent, or periodic : the series 



1 1 1 



1 + 7 + + + &c. 



11.21.2.3 



1 1 1 



1 + - + - + - + &c. 



2 3 4 



l^ - i' + .3- - 4' + &c. 

 I - 1 + 1 - 1 + &c. 



• Mutllcmnticitl Tiacls, Vul. l, p, I7II, 



3 K 2 



