Page:EB1911 - Volume 01.djvu/649

r.x^r−1, except for r ＝ 0, when x⁰ is replaced by 0. The expressions obtained in this way are called the first, second, . . . derived functions of f(x). If we denote these by f₁(x), f₂(x), . . ., so that f_n(x) is obtained from f_n−1(x) by the above process, we have

f(x+h) ＝ f(x)+f₁(x).h+f₂(x)h²/2+...+f_r(x)h^r/r+...

This is a particular case of Taylor’s theorem (see Infinitesimal Calculus).

46. Relation of Binomial Coefficients to Summation of Series.—(i.) The sum of the first n terms of an ordinary arithmetical progression (a+b), (a+2b), . . . (a+nb) is (§ 28 (i.)) 1/2n{(a+b)+(a+nb)} ＝ na+1/2n(n+1)b ＝ n_[1].a+n_[2].b. Comparing this with the table in §43 (iv.), and with formula (21), we see that the series expressing the sum may be regarded as consisting of two, viz. a+a+ . . . and b+2b+3b+ . . . ; for the first series we multiply the table (i.e. each number in the table) by a, and for the second series we multiply it by b, and the terms and their successive sums are given for the first series by the first and the second columns, and for the second series by the second and the third columns.

(ii.) In the same way, if we multiply the table by c, the sum of the first n numbers in any column is equal to the nth number in the next following column. Thus we get a formula for the sum of n terms of a series such as

2.4.6+4.6.8+..., or 6.8.10.12+8.10.12.14+...

(iii.) Suppose we have such a series as 2.5+5.8+8.11+...This cannot be summed directly by the above method. But the nth term is (3n−1)(3n+2) ＝ 18n_[2]−6n_[1]−2. The sum of n terms is therefore (§ 43 (iv.))

18n_[3]−6n_[2]−2n_[1] ＝ 3n³+6n²+n.

(iv.) Generally, let N be any rational integral function of n of degree r. Then, since n_[r] is also a rational integral function of n of degree r, we can find a coefficient c_r, not containing n, and such as to make N−c_rn_[r] contain no power of n higher than n^r−1. Proceeding in this way, we can express N in the form c_r.n_[r]+c_r−1n_[r−1]+ . . ., where c_r, c_r−1, c_r−2, . . . do not contain n; and thence we can obtain the sum of the numbers found by putting n ＝ 1, 2, 3, . . . n successively in N. These numbers constitute an arithmetical progression of the rth order.

(v.) A particular case is that of the sum 1^r+2^r+3^r + . . . + n^r, where r is a positive integer. It can be shown by the above reasoning that this can be expressed as a series of terms containing descending powers of n, the first term being n^r+1/(r+1). The most important cases are

1&nbsp+ 2 + 3 + ... + n = 1/2n(n+1),< 1²+ 2²+ 3²+ ... + n² = 1/6n(n+1)(2n+1), 1³+ 2³+ 3³+ ... + n³ = 1/4n²(n+1)² = (1+2+...+n)².

The general formula (which is established by more advanced methods) is

1/2.0^r+1^r+2^r+...+(n−1)^r ＝ 1/r+1 $\scriptstyle {\left\{{\begin{matrix}\ \end{matrix}}\right.}$n^r+1+B₁(r+1)₍₂₎n^r−1−B₂(r+1)₍₄₎n^r−3+ . . . $\scriptstyle {\left.{\begin{matrix}\ \end{matrix}}\right\}\,}$,

where B₁, B₂, . . . are certain numbers known as Bernoulli’s numbers, and the terms within the bracket, after the first, have signs alternately + and −. The values of the first ten of Bernoulli’s numbers are

B₁ ＝ 1/6, B₂ ＝ 1/30, B₃ ＝ 1/42, B₄ ＝1/30, B₅ ＝ 5/66, B₆ ＝ 691/2730, B₇ ＝ 7/6, B₈ ＝ 3617/510, B₉ ＝ 43867/798, B₁₀ ＝ 174611/330

IV. Negative Numbers and Formal Algebra.

47. Negative quantities will have arisen in various ways, e.g.

(i.) The logical result of the commutative law, applied to a succession of additions and subtractions, is to produce a negative quantity −3s. such that −3s. + 3s. ＝ 0(§ 28 (vi.)).

(ii.) Simple equations, especially equations in which the unknown quantity is an interval of time, can often only be satisfied by a negative solution (§ 33).

(iii.) In solving a quadratic equation by the method of § 38 (viii.) we may be led to a result which is apparently absurd. If, for instance, we inquire as to the time taken to reach a given height by a body thrown upwards with a given velocity, we find that the time increases as the height decreases. Graphical representation shows that there are two solutions, and that an equation X² ＝ 9a² may be taken to be satisfied not only by X ＝ 3a but also by X ＝ −3a.

48. The occurrence of negative quantities does not, however, involve the conception of negative numbers. In (iii.) of § 47, for instance, “−3a” does not mean that a is to be taken (−3.) times, but that a is to be taken 3 times, and the result treated as subtractive; i.e. −3a means −(3a), not (−3)a (cf. § 27 (i.)).

In the graphic method of representation the sign − may be taken as denoting a reversal of direction, so that, if + 3 represents a length of 3 units measured in one direction, −3 represents a length of 3 units measured in the other direction. But even so there are two distinct operations concerned in the −3, viz. the multiplication by 3 and the reversal of direction. The graphic method, therefore, does not give any direct assistance towards the conception of negative numbers as operators, though it is useful for interpreting negative quantities as results.

49. In algebraical transformations, however, such as (x−a)² ＝ x²−2ax+a², the arithmetical rule of signs enables us to combine the sign − with a number and to treat the result as a whole, subject to its own laws of operation. We see first that any operation with 4a−3b can be regarded as an operation with (+)4a+(−)3b, subject to the conditions (1) that the signs (+) and (−) obey the laws (+)(+) ＝ (+), (+)(−) ＝ (−)(+) ＝ (−), (−)(−) ＝ (+), and (2) that, when processes of multiplication are completed, a quantity is to be added or subtracted according as it has the sign (+) or (−) prefixed. We are then able to combine any number with the + or the − sign inside the bracket, and to deal with this constructed symbol according to special laws; i.e. we can replace pr or −pr by (+p)r or (−p)r, subject to the conditions that (+p) (+q) ＝ (−p)(+q) ＝ (−pq), and that + (−s) means that s is to be subtracted.

These constructed symbols may be called positive and negative coefficients; or a symbol such as (−p) may be called a negative number, in the same way that we call 2/3 a fractional number.

This increases the extent of the numbers with which we have to deal; but it enables us to reduce the number of formulae. The binomial theorem may, for instance, be stated for (x+a)ⁿ alone; the formula for (x−a)ⁿ being obtained by writing it as {x+(−)a}ⁿ or {x+(−a)}ⁿ, so that

(x−a)ⁿ ＝ xⁿ−n₍₁₎xⁿ⁻¹a+...+(−)^rn_(r)x^n−ra^r+...,

where + (−)^r means − or + according as r is odd or even.

The result of the extension is that the number or quantity represented by any symbol, such as P, may be either positive or negative. The numerical value is then represented by |P|; thus “|x|<1” means that x is between −1 and +1.

50. The use of negative coefficients leads to a difference between arithmetical division and algebraical division (by a multinomial), in that the latter may give rise to a quotient containing subtractive terms. The most important case is division by a binomial, as illustrated by the following examples:—

(1)(2)
2.10+1) 6.100+5.10+1 (3.10+1  2.10+1) 6.100+1.10−1 (3.10−1
6.100+3.106.100+3.10
2.10+1−2.10−1
2.10+1−2.10−1

In (1) the division is both arithmetical and algebraical, while in (2) it is algebraical, the quotient for arithmetical division being 2.10+9.

It may be necessary to introduce terms with zero coefficients. Thus, to divide 1 by 1+x algebraically, we may write it in the form 1+0.x+0.x²+0.x³+0.x⁴, and we then obtain

1/1+x ＝ 1+0.x+0.x²+0.x³+0.x⁴/1+x ＝ 1 − x + x² + x³ + x⁴/1+x,

where the successive terms of the quotient are obtained by a process which is purely formal.

51. If we divide the sum of x² and a² by the sum of x and a, we get a quotient x−a and remainder 2a², or a quotient a−x and remainder 2x², according to the order in which we work. Algebraical division therefore has no definite meaning unless