( 16 T )

TRACT VII.

A DISSERTATION ON THE NATURE AND VALUE OF INFINITE

SERIES.

1. About the year 1180 I discovered a very general andeasy method of valuing series, whose terms are alternatelypositive and negative, which equally applies to such series,whether they be converging, or diverging, or their terms allequal; together with several other properties relating tocertain series : and as there may be occasion to deliver someof those matters in the course of these tracts, this opportu-nity is taken of premising a few ideas and remarks, on thenature and valuation of some of the classes of series, whichform the object of those communications. This is done witha view to obviate any misconceptions that might perhaps bemade, concerning the idea annexed to the term value of suchseries in those tracts, and the sense in which it is there alwaysto be understood ; which is the more necessary, as many con-troversies have been warmly agitated concerning these mat-ters, not only of late, by some of our own countrymen, butalso by others among the ablest mathematicians in Europe,at different periods in the course of the last century ; and allthis, it seems, through the want of specifying in what sensethe term value or sum was to be understood in their disser-tations. And in this discourse, I shall follow, in a greatmeasure, the sentiments and manner of the late celebratedL. Euler, contained in a similar memoir of his in the fifthvolume of the New Petersburgh Commentaries, adding andintermixing here and there other remarks and observationsof my own.

2. By a converging series, is meant such a one whoseterms continually decrease ; and by a diverging series, that