made as small as we please by taking r large enough, so that we can make Sr approximate as closely as we please to (1+x)−m.
(iv.) To assimilate this to the binomial theorem, we extend the definition of n(r) in (1) of § 41 (i) so as to cover negative integral values of n; and we then have
| (−m)(r)＝(−m)(−m−1)...(−m−r+1)r!＝(−)rm[r] | (28), |
so that, if n ≡ −m,
| Sr ≡ 1 + n(1)x+n(2)x2+...+n(r)xr | (29). |
(v.) The further extension to fractional values (positive or negative) of n depends in the first instance on the establishment of a method of algebraical evolution which bears the same relation to arithmetical evolution (calculation of a surd) that algebraical division bears to arithmetical division. In calculating √2, for instance, we proceed as if 2.000. . . were the exact square of some number of the form c0+c1/10+c2/102+. . .
In the same way, to find X1/q, where X ≡ 1+a1x+a2x2+... and q is a positive integer, we assume that X1/q＝1+b1x+b2x2..., and we then (cf. § 55) determine b1, b2, . . . in succession so that (1+b1x+ b2x2+. . .)q shall be identical with X.
The application of the method to the calculation of (1+x)n, when n＝p/q, q being a positive integer and p a positive or negative integer, involves, as in the case where n is a negative integer, the separate consideration of the form of the coefficients b1, b2, . . . and of the numerical value of 1 + b1x+b2x2+. . .+brxr.
(vi.) The definition of n(r), which has already been extended in (iv.) above, has to be further extended so as to cover fractional values of n, positive or negative. Certain relations still hold, the most important being (22) of § 44 (ii.), which holds whatever the values of m and of n may be; r, of course, being a positive integer. This may be proved either by induction or by the method of § 52 (vi.). The relation, when written in the form (23), is known as Vandermonde's theorem. By means of this theorem it can be shown that, whatever the value of n may be,
{1+(p/q)(1)x+(p/q)(2)x2+...+(p/q)(r)xr}q＝1+p(1)x+p(2)x2+...+p(r)xr+terms in xr+1, xr+2,...xqr.
(vii.) The comparison of the numerical value of 1+n(1)x+n(2)x2+ ... +n(r)xr, when n is fractional, with that of (1+x)n, involves advanced methods (§ 64). It is found that this expression can be used for approximating to the value of (1+x)n, provided that |x| < 1; the results are as follows, where ur denotes n(r)xr and Sr, denotes u0+u1+u2+. . .+ur.
(a) If n > −1, then, provided r > n,
(1) If 1 > x > 0, (1+x)n lies between Sr and Sr+1;
(2) If 0 > x > −1 , (1+x)n lies between Sr and Sr +ur+1/(1+x).
(b) If n < −1, the successive terms will either constantly decrease (numerically) from the beginning or else increase up to a greatest term (or two equal consecutive greatest terms) and then constantly decrease. If Sr is taken so as to include the greatest term (or terms), then,
(1) If 1 > x > 0, (1+x)n lies between Sr and Sr+1;
(2) If 0 > x > −1, (1+x)n lies between Sr and Sr + ur+1/(1−ur+1/ur).
The results in (b) apply also if n is a negative integer.
(viii) In applying the theorem to concrete cases, conversion of a number into a continued fraction is often useful. Suppose, for instance, that we require to calculate (23/13)32. We want to express (23/13)3 in the form a2b, where b is nearly equal to 1. We find that 32 log10 (23/13)＝.3716767＝log10(2.3533)＝log10(40/17) nearly; and thence that (23/13)32＝(40/17) (1+1063/3515200)12, which can be calculated without difficulty to a large number of significant figures.
(ix.) The extension of n(r), and therefore of n[r], to negative and fractional values of n, enables us to extend the applicability of the binomial coefficients to the summation of series (§ 46 (ii)). Thus the nth term of the series 2⋅5+5⋅8+8⋅11+...in § 46 (iii.) is 18(n−13)[2]; formula (20) of § 43 (iv.) holds for the extended coefficients, and therefore the sum of n terms of this series is 18⋅(n−13)[3]−18⋅(−13)[3]＝3n3+6n2+n. In this way we get the general rule that, to find the sum of n terms of a series, the rth term of which is (a+rb)(a+r+1⋅b)...(a+r+p−1⋅b), we divide the product of the p+1 factors which occur either in the nth or in the (n+1)th term by p+1, and by the common difference of the factors, and add to a constant, whose value is found by putting n＝0.
57. Generating Functions.—The series 1−m[1]x+m[2]x2— . . . obtained by dividing 1+0⋅x+0⋅x2+ . . . by (1+x)n, or the series 1+(p/q)(1)x+(p/q)(2)x2 + . . . obtained by taking the qth root of 1+p(1)x+p(2)x2+ . . . , is an infinite series, i.e. a series whose successive terms correspond to the numbers 1, 2, 3, . . . It is often convenient, as in § 56 (ii.) and (vi.), to consider the mode of development of such a series, without regard to arithmetical calculation; i.e. to consider the relations between the coefficients of powers of x, rather than the values of the terms themselves. From this point of view, the function which, by algebraical operations on 1+0⋅x+0⋅x2+ . . . , produces the series, is called its generating function. The generating functions of the two series, mentioned above, for example, are (1+x)−m and (1+x)p/q. In the same way, the generating function of the series 1+2x+x2+0⋅x3+0⋅x4+. . . is (1+x)2.
Considered in this way, the relations between the coefficients of the powers of x in a series may sometimes be expressed by a formal equality involving the series as a whole. Thus (4) of § 41 (ii.) may be written in the form
1 + n(1)x+n(2)x2+...+n(r)xr+...=f (1+x){1+(n−1)(1)x+(n−1)(2)x2+...+(n−1)(r)xr+...};
the symbol “ =f ” being used to indicate that the equality is only formal, not arithmetical.
This accounts for the fact that the same table of binomial coefficients serves for the expansions of positive powers of 1 + x and of negative powers of 1 − x. For (4) may (§ 43 (iv.)) be written
(n − 1)[r]＝n[r]−n[r−1],
and this leads to relations of the form
| 1+2x+3x2+... =f (1−x)(1+3x+6x2+10x3+...) | (30), |
each set of coefficients being the numbers in a downward diagonal of the table. In the same way (21) of § 43 (iv.) leads to such relations as
| 1+3x+6x2+... =f (1+x+x2+...)(1+2x+3x2+...) | (31), |
the relation of which to (30) is obvious.
An application of the method is to the summation of a recurring series, i.e. a series c0+c1x+c2x2+...whose coefficients are connected by a relation of the form p0cr+p1cr−1+...+pkcr−k＝0, where p0, p1, . . . pk are independent of x and of r.
58. Approach to a Limit.—There are two kinds of approach to a limit, which may be illustrated by the series forming the expansion of (x+h)n, where n is a negative integer and 1 > h/x > 0.
(i) Denote n(r)xn−rhr by ur, and u0+u1+ . . . +ur by Sr. Then (§ 56 (iii.)) (x+h)n lies between Sr and Sr+1; and provided Sr includes the numerically greatest term, |Sr+1−Sr| constantly decreases as r increases, and can be made as small as we please by taking r large enough. Thus by taking r＝0, 1, 2, . . . we have a sequence S0, S1, S2, . . . (i.e. a succession of numbers corresponding to the numbers 1, 2, 3, . . .) which possesses the property that, by starting far enough in the sequence, the range of variation of all subsequent terms can be made as small as we please, but (x+h)n always lies between the two values determining the range. This is expressed by saying that the sequence converges to (x+h)n as its limit; it may be stated concisely in any of the three ways,
(x+h)n＝lim(xn+n(1)xn−1h+...+n(r)xn−rhr+...), (x+h)n＝lim Sr, Sr ≐ (x+h)n.
It will be noticed that, although the differences between successive terms of the sequence will ultimately become indefinitely small, there will always be intermediate numbers that do not occur in the sequence. The approach to the limit will therefore be by a series of jumps, each of which, however small, will be finite; i.e. the approach will be discontinuous.
(ii) Instead of examining what happens as r increases, let us examine what happens as h/x decreases, r remaining unaltered. Denote h/x by θ, where 1 > θ > 0; and suppose further that θ < |1/n|, so that the first term of the series u0+u1+u2+ . . . is
