(C). 1 x 1 1 we x as the first The process by which we have obtained (C) is clearly the same thing as to divide 1 by x + 1, using term of the divisor. Thus, if we divide 1 by Rule I., we have + 1, by consequently, we easily obtain (C), as given above. If x is sensibly greater than 1, so that x1 is a finite quantity, it is clear that the integer n may be taken so great shall become an insensible quantity, which that 1 x(x-1) may be rejected; which being done, (C) is reduced to 1 x = 1 1 [++ + etc., ], (D), which is an infinite series, whose law of formation is evident. It is clear that the series in (D) will enable us to calculate the generating function 1 1 - x to any required degree of ex actness when a 1 is not a very small quantity. = − 0.032258, etc., so that three terms of the series give the correct value of the generating function to five decimal places. If, however, -1 is an indefinitely small quantity, the series will not serve to calculate the numerical value of the generating function, since it is indefinitely great. It follows, from what has been done, that if in any calculation we meet with the series 1 + x + x2 +x3 + etc., to infinity, such that x is sensibly less than 1, then we may put 1 for it. But if 1 is an indefinitely small positive quantity, then the series will indicate that the generating function is infinitely great. If is sensibly greater than 1, we must put the series 1 1 1 + + + etc., to infinity], for the given series; because the generating function has been expanded according to the ascending powers of x, when it ought to have been. expanded according to the descending powers of x, or in the form · [x-1 + x2 + x-3+ etc.], (which are the descending powers, since they decrease by subtraction), and — [x-1 + finity, however great the shall have in all cases 1 1 x Hence, we = 1 + x + x2 + 23 + etc., to in finity (E), where it is to be observed if x is sensibly greater than 1, that (=) the sign of equality must not be understood to mean that the numerical value of the series is the same as 1 1 that of but only that it is the expansion of ac 1 1 cording to the ascending powers of x, when it ought to have been the expansion according to the descending powers of x. What is here said does not prevent our using 1 + x + + + etc., in calculation (when x is greater than 1), since we can put the series in (D) for it; or, which is the same, we may always put the generating function 1 x for it. It may be shown, in a similar way, if we meet with the It is evident that similar observations must always have place, when we convert any fractional expression into an infinite series. It may be added that the great use of a series is to compute the value of the quantity of which it is the expansion, or which is the same, the generating function of the series; consequently, the series must be such that its numerical value shall equal that of the generating function when we reject indefinitely small quantities, so that we see the reasonableness of the preceding observations. Ex. 2.-Divide 1 by 1 + x. If we take 1 for the first term of the divisor, then we proceed as follows. x + x2 xn + (A); 1 + x2 we must be 1 + x observing that n is a positive integer, and that for must use + if n is an even number, and that used for when n is an odd number. If a is sensibly less than 1, then n may be taken so great = 1−x + x2 - may be rejected, and we shall have 1 1 + x x2 + x1 — 2+ etc., (B), the law of continuation of the series being such that if we multiply any term by-x, the product will be the next successive term. xn If a equals 1, or differs insensibly from 1, then ± be 1 1 + x comes sensibly ± and of course can not be rejected how 2' ever great n may be, so long as it is finite; and if we regard zas absolutely equal to 1, we can not reject ± Because in (B) is less than 1, we shall have a greater than xx= x2, and a greater than xxx3, and so on; and since the terms are alternately + and -, 1 x is less than the value of the whole series, since we must add the positive quantity 203 + 20$ — 205 + etc., to it to make up the series; also 1 23 is less than the value of the whole x + x2 series, since we must add the positive quantity — x + xổ x6 - etc. to it, to complete the series; and in this way we see that the sum of any even number of terms is less than the true value of the series. Also 1 is greater than the value of the series, since we must subtract the positive quantity x — x2 + x3 — etc. from 1, to get the series; and in the same way 1 x + x2 is greater than the value of the whole series, since we must subtract the positive quantity a ++ etc., from 1-x+ to get the series; and in this manner it appears that the sum of any odd number of the successive terms of (B) is greater than the value of the whole series. From what has been done, it follows that 1 1 + x may be considered as having the sums of an even and odd number of the successive terms of the series (B) for certain limiting values or limits, since is greater than the sum of the even number, and less than the sum of the odd number of terms. 1 Hence, if we add any number of the terms of the series, and use the integer m to stand for the number of terms add1 ed, it is clear that the difference between and the sum 1 + x of m terms of the series will be numerically less than the (m + 1) term of the series; since the sums of m and (m + 1) terms of the series are limits to the value of +1 1 + x It is clear that what has been said is applicable to any converging series whose terms are alternately and —. We will now convert 1 1 + x the first term of the divisor; into a series by taking a for (C); observing that n is a positive integer, and that for ± we must use when n is an even number, but if n is an odd number, then for we must use +. If a is sensibly greater than 1, we may take n so great that may be rejected, and (C) will be reduced to 1 x" (1 + x) |