TRACT 7.

OF INFINITE SERIES.

175

that, in series whose terms do not converge to 0, we do notconnect that notion with this idea of a sum, namely, that themore terms of the series are actually collected, the nearer wemust approach to the value of the sum.

14. But if any person shall still think it improper to applythe term sum, to the finite expressions by whose evolutionall series in general are produced ; it will make no differencein the nature of the thing ; and instead of the word sum, forsuch finite expression, he may use the term value, or func-tion, or perhaps the term radix would be as proper as anyother that could be employed for this purpose, as the seriesmay justly be considered as issuing or growing out of it, likeas a plant springs from its root, or from its seed. The choiceof terms being in a great measure arbitrary, every person isat liberty to employ them in whatever sense he may thinkfit, or proper for the purpose in hand ; provided always thathe fix and determine the sense in which he understands oremploys them. And as I consider any series, and the finiteexpression by whose evolution that series may be produced,as no more than two different ways of expressing one and thesame thing, whether that finite expression be called the sum,or value, or function, or radix of the series ; so in the follow-ing paper, and in some others which may perhaps hereafterbe produced, it is in this sense I desire to be understood,when searching out the value of series, namely, that the ob-ject of the enquiry, is the radix by whose evolution the seriesmay be produced, or else an approximation to the value ofit in decimal numbers, &c.