Before, however, we proceed to reason upon them, we must distinctly understand what we mean by an infinite series. Some persons cannot an infinite series, except by means of successions of finite thus they have no other idea of 1+ 2+ 4 + 8+ ..., except as of which the conception is a pure result of the successive consideration of 1, 1+2, +2+4, 1 + 2+ 4+ 8, &c. If they can no further than this, that is, if at no of their contemplation can they treat I +2+ 4 + an more than carried to some enormous number of terms, with a to carry it further ; we can then concede to them the to object, in the manner described, to the use of a divergent series, we think it possible that even in this case an answer be to the objection. But if there be any who can with us carry their notions further, and treat the series as absolutely endless, in the same manner as we are to conceive time and space to be absolutely endless, upon the result not as to its arithmetical value, but as to its form and capability of being the object of operations—we then think that we have those with whom the of series can be on more like a basis of demonstration. They may arrive at the final idea by means of the successions which the first class of thinkers say must end somewhere ; but they answer, that this is no more true than that space must end somewhere : if it be that we are capable of a line extended without limit, with parts set off its total infinite it must equally be granted that we might suppose one term of a series written at each and every point of sub division. To this issue the be namely, the alternative of the conception of the infinite series, or of that of the infinite line. And it must be remarked that the considerations by which we limited the use of the word INFINITE in that article do not apply hero, for we are not upon any supposed* attainment of the other end of the line, but upon ideas derived from a process of successions carried on such attempt as we can make in out towards that attainment.
This being premised, let us now consider the series 1 + a .
ad infinitum, the last words being used strictly in the above sense, without reference to any particular value of a, and only as an object of algebraical operation. To what finite function of a is this an algebraical equivalent in all matters of operation ? Let us consider first merely results of operation, without any question as to whether the series operated on have or not, or whether expressions which appear to be the same so far as operations are concerned, are to be the same in value or not, when any difficulty arises as to the value of either. We assume those five rules of operation and their consequences, on which [OPERATIoN] the technical part of algebra is founded. If we then call the preceding series P, we find that P and 1 +a P are the same series. If then r=1 +a P, we find e=1 : (1—a), a result which is certainly not true in any arithmetical point of view, when a>1 ; for in such a case the series is infinite, and the finite expression negative. Leaving this, let us assume, for trial if the reader pleases, the equation 1 : (1— a)= s„a• ; in this change a into I : a, and add, which gives I a = 2 + + + 84a—' or 1+ 8, (a' + 0, a result which is again perfectly incongruous in an arithmetical point of view. At full length it is
1 1 I + a + + a 2 + + . . . . = 0 .
To test this curious result, by operations merely, call it Oa, and multiply A + na + .... by it : the result, by common rules, will be found to bo (s.+B+c+ ) = (-t + + • • • .) ; a result which agrees perfectly with epa = 0, and with no other sup position whatsoever.
A great many other instances might bo given, in which the use of (pa = 0 makes sense, so to speak, of results in the formation of which cpa has been used. And it is generally admitted that divergent series are found to make sense, in the same manner, of almost every result in the formation of which they are used ; and also that when such results happen themselves to be free of divergency, there is very rarely any distinction, as to either truth or clearness, between them and the results of ordinary algebra ; insomuch that the objection of those who would avoid them altogether, as usually stated, amounts, so far as operations are concerned, to the assertion that they sometimes give false results.
If we then compare the position in which we stand with respect to divergent series, with that in which we stood a few years ago with respect to impossible quantities, we shall find a perfect similarity. The divergent aeries, that is, the equality between it and a finite expression, is perfectly incomprehensible in an arithmetical point of view ; and so was the impossible quantity. The use of divergent series has been admitted, by ono on one explanation, and by another on another, almost ever since the commencement of modern algebra ; and so it was with the impossible quantity. It became notorious that such use generally led to true results, with now and then an apparent exception, which most frequently ceased to he such on further consideration ; this is well known to have happened with impossible quantities. In both cases these apparent exceptions led some to deny the validity of the method which gave rine to them, while all were obliged to place them both among those parts of mathematics (once more extensive than now) in which the power of producing results had outrun that of interpreting them. But at larst came the complete explanation of the impossible quantity [Atcenne], showing thatall the difficulty had arisen from too great limitation of definitions ; and almost about the same time arose that disposition among the con tinental writers, of which we have spoken above, namely, to wait no longer for the explanation of the true meaning of a divergent series, but to abandon it altogether. But why should the divergent series, of all the results of algebra which demand interpretation, be the only • one to be thrown away without further inquiry, when in every other Case patience and research have brought light out of darkness. So far as the matter has yet gone, very little has been done towards the interpretation of a divergent series independently of its invelop ment, or function from which it is developed. When this invelop ment is known, and the series deduced from it, there are means of stopping the divergency, by arresting the development at any given point, and turning the remainder, not into a further development, but into a finite form. Thus if cps, a given function of x, should givo divergent series A„ + e,x + , all that part of the development which follows A. x^ may be included in (x—r) ride : 1.2.3 n.