application, it has usually to be ascribed solely to the circumstance that the calculation has been conducted as if the several terms of the series had been still multiplied by the successive powers of an expression x, which does not else appear in the terms themselves, that is, as if they were series which proceeded according to whole powers of x*. If on the other hand we consider all the terms in such an infinite series which no longer proceeds according to whole powers of any expression x, as determinate values in cyphers, i. e. as perfectly determinate actual or imaginary numbers of the form p+q-1, these infinite series may be called numerical series in contradistinction to the general series, and in these numerical series we must carefully distinguish two cases. (a) If the result obtained by adding together n consecutive terms of the series, receives a perfectly determinate actual or imaginary (limiting) value p+q-1+ for n= (i. e. when n is infinitely great, i. e. in case a positive whole number, greater than any determinate number however great, is substituted for n); in this case the (numerical) series is termed convergent (and we say it converges) and the above mentioned actual or imaginary number p+q√-1, which has resulted from putting n = ∞ in the result of the addition of the n first consecutive terms of the series, is then called the value of the convergent (numerical) series; (b) If the addition of n terms of the series gives an expression of the form P2+Q-1, in which however one or each of the functions P or Q2 becomes, for n = ∞, itself infinitely great (i. e. positive or negative, but absolutely greater than any determinate whole or broken number however great); such a (numerical) series is called divergent (and we say it diverges). As long, therefore, as an infinite series is still conceived as general, we cannot speak about its divergence or convergence, precisely because, according to what has gone before, these latter ideas can only make their appearance in conjunction with numerical series (that is, when the general series are expressly considered as numerical). From these ideas the following most important conclusions may now be drawn: *And herein we see the explanation of the circumstance that "calculation" with divergent numerical series, as may be often found in the writings of the mathematicians of the last century, may nevertheless lead to a correct result. + The result of the addition of n such terms has namely the form P+Q. √−1, where P and Q are functions of n which receive actual values for every positive whole n, while Q may also be 0. If then P and Q receive determinate actual values Р and q for n∞, the above case occurs. The numbers p and q may however be only given by finding that P, and Q lie for n = ∞ between two limits which may be made to approach one another as near as we choose, in other words, p and g may be irrational. What is here called value is frequently termed by other writers sum. But we here distinguish accurately between value and sum, and only use the word sum in the signification attached to this word in (sect. 47). (1) The convergent numerical series has a "value" which is actual or imaginary and of the form p+q√-1; this "value" can always be substituted for it in all " calculations," since it is "equal" to it. And inasmuch as it always reproduces this "value" and no other than this value, we can always "calculate” with numerical series which are also convergent. (2) Now a convergent series has this value only by virtue of the law according to which its terms proceed in infinitum. Hence to prevent any doubt occurring concerning the true value of a convergent series, we must carefully enunciate the law according to which the terms are to be taken in infinitum. For example, from the same reciprocal terms of the natural numbers, viz. 1 1 1 7' we may compose any number of numerical infinite series, which are all convergent but have all different values, each however having its own perfectly determinate value by virtue of the determinate law according to which it is constructed. in which, if we take the first 3n terms, 2n positive and only n negative of the above terms follow one another (in their order) has for its value log. nat. 2. in which, if we take the first 3n terms, we find n of them positive and 2 n negative, has for its value log. nat. 2. And if we take μ of the above terms positive and v negative, and then μ positive and negative, and so on, the value of the resulting numerical convergent series is log. nat. 2 + log. nat., and therefore = log. nat. 2, which is that of the first series, when м when μ<v. v, but greater than that when μμ> v and less (3) A divergent series has no value which it can represent; a divergent (numerical) series is therefore a form inadmissible in calculation, exactly as the form must be acknowledged as such. b 0 was above shewn to be, and Consequently, if a general series becomes a numerical one, in any particular case of application, and this numerical series is found to be divergent, we are instantly incapable of calculating with it, and the result, without being incorrect, shews decisively, that the general calculation previously employed no longer holds when the letters receive these values in cyphers, and that the calcu lation must be recommenced and particularly conducted for this particular case* * *Among divergent series are those in which the same terms recur periodically in infinitum, as the series 1 − 3 + } − 1 + 1 −}+}−}+1− }+-+1− in inf. (4) From every numerical series R, whether convergent or divergent, a general series can be formed, by subjoining to the several terms the different powers of a general expression x, as factors. If the new series be designated by R., the numerical series R, will result from putting x = 1 in the general series R. Now suppose the general series R, to have a "sum" (in the sense of (sect. 47) which may be represented by S, so that we have (O). . . . . . . . . R2 = S2, = and that this sum S, receives for x 1 a determinate value in cyphers, which may be represented by S,, then the equation (O) becomes for x = 1, Now if the series R, is convergent, then its "value" is necessarily S1, but if the series R, is divergent, it has no value, and that, which has no existence, cannot of course pretend to be equal to the value S1. The equation R1 = S, ceases to exist, as far as calculation is concerned, when the series R, is divergent, just because the divergence of the series always shews by (No. 3), that the general calculation here suffers an exception. This may be now more generally enunciated. If a general series R, has generally the "sum" S,, so that the equation (a). . . . . . . . . R2 = Sr is a correct one (in the sense of sect. 3); and if a determinate value in cyphers is given to x, so that the series R. becomes a numerical series R, while S, receives the value in cyphers S, the equation (6)...... R = S ......... is still correct, provided R is convergent, i. e. S is then the "value" of the infinite convergent series R; while if the numerical series R is divergent, the equation (6) does not enunciate anything which is incorrect, but it no longer holds; it ceases to exist, and can no longer be regarded in calculation*. And still more generally. If two forms R, and S., which are either both finite or of which one contains infinite series, or both contain infinite series, are equal to one another, and if R and S represent those forms expressed in cyphers, which result from substituting a determinate value in cyphers for x in R, and S respectively, then the equation RS is correct, as long as R and S have determinate "values" (in cyphers); and on the other hand not incorrect, but no longer admissible in calculation, when a *This is precisely the same case as that of the equation .a=b. This equation is always correct as long as a is general (that is, a mere supporter of the operations);-but if the value 0 (zero) is given to a, it does not become incorrect, but entirely ceases to exist, because a form like cannot be admitted in calculation, It always points out an exception. ever one or each of the forms in cyphers R and S, contains a (numerical and) divergent infinite series. Since these doctrines and rules which result necessarily from the present views, have been here and there overlooked and disregarded by analysts, we will illustrate the subjects we have just been treating, by a few examples. Thus, for example, the "sum" of the general series value of x, so that its value for x=1, for example, is necessarily . But for every value of x>, as e. g. for x = 1, this same infinite series diverges, that is, has no value at all; and that which has no existence cannot of course be found from the is therefore a perfectly correct equation. Both forms to the right and left of the sign of equality have the one single property which the quotient on the right represents, viz. that when they are multiplied by 1+ 2x the result is exactly = 1. Either general expression may therefore be unconditionally substituted for the other in all "calculations." And in the case that both expressions to the right and left of the sign of equality receive values in cyphers, these two values have the same property and are therefore still "equal" to one another. But we are of course unable to say any thing at all about this latter circumstance, when one of the two values no longer exists, i. e. when the series diverges. We may also transform the quotient 1 1 + x series proceeding according to whole powers of or 1 x 1 x + 1 into a We have If we now transpose in this last equation some of the terms to the right from the left, or to the left from the right, we have the additional correct equations: 1 1 1 1 1 (4) ¦ ¦ − ¦¦ + 1 − x + x2 - a2 + in inf. -- +- + in inf. Ꮖ x3 If we now write the left side of the equation (4) thus -in inf. If we write in equation (5) the expression on the right thus, and these two last equations are also unconditionally correct (by sect. 3). But if we now give determinate values in cyphers to x, then the infinite series are no longer general, but numerical, and the equations without becoming incorrect, either cease to exist, or become correct equations in cyphers. (No. 1) viz. gives on the left the value of the numerical convergent series on the left when a <1; when x>1 the equation (No. 1) ceases to exist, because the series then diverges. The first is the case in (No. 2) when x>1, the second when x<1. The equation (No. 3) although it is entirely and perfectly correct, and although we have deduced from it the perfectly correct equations (Nos. 4, 5, 6, and 7) nevertheless always ceases to exist when any value in cyphers is substituted for x, because any value of x which would make one side of the equation a convergent series, would always make the other a divergent series. The same is also true for the equations (Nos. 4 and 5); they do not exist for any one value in cyphers of x. But nevertheless, the generally correct equation (No. 6) which is deduced from (No. 4), does exist again for every value in cyphers of x, which is>í, and ceases to exist when x<1. Similarly for the equation (No. 7) which has been deduced from the equation (No. 5) that does not exist for any single value in cyphers of x, and which is as generally correct as all the others, and which also exists for every value in cyphers of a that is <1, but ceases to exist whenever x>1. Let us now take, in the second place, an example in which a root occurs. Suppose, for example, we have to extract the m root of the series |