SERIES, infinite, is a series consisting of an infinite number of terms, that is, to the end of which it is impossible ever to come; so that let the series be carried on to any assignable length, or number of terms, it can be carried yet further, with out end, or limitation.
A number actually infinite, (that is, all whose nits can be actually assigned, and yet is without limits,) is a plain contradic tion to all our ideas about numbers; for whatever number we can conceive, or have any proper idea of, is always deter minate, and finite ; so that a greater after it may be assigned, and a greater after this; and so on, without a possibility of ever coming to an end of the addition or increase of numbers assignable ; which inexhaustibility, or endless progression, in the nature of numbers, is all we can distinctly understand by the infinity of number; and therefore to say, that the number of any things is infinite, is not saying that we comprehend their num ber, but indeed the contrary ; the only thing positive in this proposition being this, that the number of these things is greater than any number which we can actually conceive and assign. But then, whether in things that do really exist, it can be truly said that their number is greater than any assignable number ; or, which is the same thing, that in the nu meration of their units, one after another, it is impossible ever to come to an end ; this is a question about which there are different opinions, with which we have no business in this place ; for all that we are concerned here to know is this certain truth, that, after one determinate number we can conceive a greater, and after this a greater, and so on without end. And, therefore, whether the number of any things that do or can really exist all at Once can be such, that it exceeds any de terminable number, or not, this is true, that of things which exist, or are produc ed successively one after another, the number may be greater than any assign able one; because, though the number of things thus produced, that does actually exist at any time, is finite, yet it may be increased without end. And this is the distinct and true notion of the infinity of a series, that is, of the infinity of the MM.
ber of its terms, as it is expressed in the definition.
Hence it is plain that we cannot apply to an infinite series the common notion of a sum, viz. a collection of several particu lar numbers that are joined and added together one after another ; for this sup poses that these particulars are all known and determined ; whereas the terms of an infinite series cannot be all separately as signed, there being no end in the nume ration of its parts, and therefore it can have no sum in sense. But again, if we consider that the idea of an infinite series consists of two parts, viz. the idea of something positive and determined, in so far as we conceive the series to be actu ally carried on : and the idea of an inex haustible remainder still behind, or an endless addition of terms that can be made to it one after another, which is as different from the idea of a finite series as two things can be : hence we may con ceive it as a whole of its own kind, which, therefore, may be said to have a total value, whether that be determinable or not. Now in some infinite series this value is finite or limited ; that is, a num• ber is assignable, beyond which the sum of no assignable number of terms of the series can ever reach, nor indeed ever be equal to it, yet it may approach to it in such a manner as to want less than any assignable difference ; and this we may call the value or sum of the series ; not as being a number found by the common method of addition, but as being such a limitation of the value of the series, taken in all its infinite capacity, that if it were possible to add them all, one after Another, the sum would be equal to this number.
Again, in other series the value has no limitation; and we may express this, by saying the sum of the series is infinitely great : which, indeed, signifies no more than that it has no determinate and as signable value ; and that the series may be carried such a length as its sum, so far, shall be greater than any given num ber. In short, in the first case, we affirm there is a sum, yet not a sum taken in the common sense ; in the other case, we plainly deny a determinate sum in any sense.