REPORT ON CERTAIN BRANCHES OF ANALYSIS. 273 



degree of accuracy, to its value by the aggregation of a finite 

 number of those terms. Many tests of the summability of series 

 (considered as different from the determination of their gene- 

 rating functions,) have been proposed, possessing very diflterent 

 degrees of certainty and applicabiUty. The geometrical series 

 which we have just been considering is convergent or divergent, 

 that is, summahle or not, according as x is greater or less than 1 ; 

 and it is convenient, for this and for other reasons, to assume it 

 as the measure of the convergency or divergency of other series. 

 If it can be shown that a converging geometrical series can be 

 formed whose terms within a finite and assignable distance from 

 the origin become severally greater than those corresponding 

 to them of the assigned series, then that series is convergent. 

 And if it can be shown that a divergent geometric series can be 

 formed whose terms within a finite and assignable distance from 

 the origin are severally less than those corresponding to them 

 of the assigned series, then that series is divergent*. Such 

 tests are certain, as far as they are applicable ; but there may 

 be many cases, both of divergent and convergent series, which 

 they are not sufficiently delicate to comprehend. 



It would appear from the preceding observations that di- 

 verging series have no arithmetical sums, and consequently 



* Peacock's Algebra, Art. 324, and following. Caucliy, Cours d'jinalyse 

 Algebrique, chap. vi. This last work contains the most complete examination 

 of the tests of convergency with which I am acquainted. 



The measure of convergency mentioned in the text, which was first sug- 

 gested and applied by D'Alembert, will immediately lead to the following: 



*• If u represent the n^^ term of a series, it is convergent (or will become so) 

 j_ 

 if the superior limit of ("„)" be less than 1, when n is infinite; divergent in 

 the contrary case." 



"If the limit of the ratio w„ , j to u^ be less than 1, the series is convergent, 

 and divergent in the contrary case." 



Many other consequences of these and other tests are mentioned by Cauchy 

 in the work above referred to. 



M. Louis Olivier, in the second volume of Crelle's Journal, has proposed the 

 following test of convergency. " If the limit of the value of the product n u^ 

 he finite or zero when n is infinite, then the series is divergent in the first case, 

 and convergent in the second." This principle, however, though apparently 

 very simple and elementary, has been shown by Abel, in the same Journal, to 

 be not universally true. Thus, the series 



111 1 



2 log 2 3 log 3 ^ 4 log 4 • • • • ^ n\ogn 

 may be shown to be infinite, though the product n v„ is equal to zero when n is 

 infinite. The same acute and original analyst has shown that there is no func- 

 tion of n whatever which multiplied into v^ will produce a result which is zero 

 or finite when n is infinite, according as the series is convergent or divergent. 



1833. T 



