·(4),

it will only remain to determine the values of the coefficients A,, A2, A3, &c. and to show the law of their dependence on the index (m) of the operation by which they are produced.

444. For this purpose, let m denote any number whatever, whole or fractional, positive or negative; and for, in the above formula, put y+z; then,

there will arise (1+)"=[1+(y+z)]"=[(1+y)+

z]m, which being all identical expressions, their expansions, when taken according to the above form, will evidently be equal to each other.

445. Whence, as the numeral coefficients A1, Ag, A3, &c. of the developed formula, will not change for any value that can be given to a and x, provided the index (m), remains the same, the two latter may be exhibited under the forms

[! + ( 1+z)]"=1+A¿(y+z)+A2(y+z)3 + &c. [(1+y)+z]TM=(1+y)TM+A,z(1+y)TM-'+A2z2 (1+y)

--2+ &c.

And, consequently, by raising the several terms of the first of these series to their proper powers, and putting 1+y=p in the latter, we shall shall have 1+A, (y+2) +A, (y2+2yz+z3) +A ̧(y3+3y3z+ 3yz3 +23)+&c.=pm+A1p" + &c.

m--1

m--22

3

446. Or, by ordering the terms, so that those which are affected with the same power of z may be all brought together, and arranged under the same head, this last expression will stand thus:

+3A,y +4A1y A,y3 +3A ̧y2 +6A‚y3 | +10A ̧ya A ̧y3+4A ̧y3 +10Asy3 +30Ay

3

&c.

&c.

3

&c.

≈3+ &c. (5).

In which equation it is evient, that both y and zare indeterminate, and independent of the values of A,, A,, A., &c.; since the result here obtained arises solely from the substitution of the sum of these quan

tities for in equation (4).

a

447. Hence, as the first terms and the coefficients, or multipliers of the like powers of 2, in these two expressions, are, in this case, identical, (Arf. 435), we shall have, by comparing the first column of the lefthand member with the first term of that on the right, :+A1y+A ̧y2+A ̧y3 +A1y* + &c• = p. which is an indentity that verifies itself; since, by hypothesis, (1+)m=pm, and, according to the ral formula, (+y)=1+A1y+A,y3+A ̧y3+&c. 448. Also, if the second of these columns be compared in like manner, with the second on the right, there will arise the new identity,

gene

A, +2A2y+3A,y2+4A1y3=A,pm; which will be sufficient, independently of the rest of the terms for determining the values of the coefficients A1, A ̧, A1, &c.

3

3

For since Ap1=A,

p.

A

·(1+A‚y+A ̧y2+A ̧

y3+ &c.), the equating this series with the last, and multiplying the left-hand side by 1+y. will give [A + 2A y + 3A ̧ y2+ &c.](1+y)=Â+A‚Ä‚y+A, à ̧y3 +A‚ ̧у3 + &c.

3

And, therefore, by actually performing the operation, and arranging the terms accordingly, we shall Trave

A,+2A, y+3A | y2+4A, | y2+ &c.

3

449. From which last identity, there will obviously arise, by equating the homologous terms of its two members, the following relations of the coef ficients :

And, consequently, as the coefficient A, of the second term of the expanded binomial, has been shown to be equal, in all cases, to the index (m) of the proposed binomial, the last of these expressions will become of the form

where the law of the continuation of the terms, from A, to the general term A, is sufficiently evident.

450. Whence it follows, that, whether the index m be integral or fractional, positive or negative, the proposed binomial (a+x)", when expanded, may always be exhibited under the form

"(1+2)·

m(m—1) (x), m(m −1).(m—2) +.

m m 2

2.3

Ꮖ

And if — be substituted in the place of +2, the

-

α

same formula will, in that case, be expressed as fol

m(m − 1). (m—2) ( : ) 3 +&c.];

2.3

or (a-x)-a-mam-1x+

m(m-1). (m2) am-3x3 &c.

2.4

Where it is to be observed, that the series, in each of these cases, will terminate at the (m+1)th term, when m is a whole positive number; but if m be. fractional or negative, it will proceed ad infinitum ; as neither the factors m-1, m−2, m-3, &c. can then become =0.

451. To this we may add, that in the two last instances here mentioned, the second term

()

of the

binomial must be less than 1, or otherwise the series, after a certain number of terms, will diverge, instead of converging.

452. It may also be farther remarked, that when a and x in these formulæ, are each equal to 1, we shall have, agreeably to such a substitution, (a+x)m= m(m−1) ̧m (m—1). (m—2) (1 +1)"=2′′=1+m+! +

2

+ &c., and

2.3

m

(α-x)=(1-1)"=0"=0=1-m+
m(m—1).(m—2) ̧ (m−1)m−2). (m-3)

2.3

+

2.3.4

From which it appears, that the sum of the coefficients arising out of the developement of the mth power, or root of any binomial, is equal to 2m; and that the sum of the coefficients of the odd terms of the nth power, or root of a residual quantity, is equal to the sum of the coefficients of the even terms.

453. Finally, let m=0; then (a+x) =a to

&c. where it i evident that the series terminates at the first term (); since the coefficient of every successive term involves 0 for one of its factors; therefore (a+ra°=i, (Art. 86). And, if a=x; then (a—x)° = a' = 1. that is, 0°=1. Hence, it follows, that any quantity, either simple or compound, raised to the power 0 is equal to unity or 1; and also that 0° is, in all cases, equal to unity or 1.

454. Although it has been observed (Art. 167), that 0° appears to admit of an infinity of numerical values; because it is equal to %, which is the mark of indetermination; yet it is plain, from what is above shown, that 0° is only one of the values of 8, which,

in that particular case (Art. 167), where

Om

0m

=

0

is equal to unity. The intelligent reader is referred to BONNYCASTLE'S Algebra, 8vo. vol. ii. Also, LAGRANGE'S Theorie des Fonctions Analytiques, and Leçons sur le Calcul des Fonctions.

§ H. APPLICATION OF THE BINOMIAL THEOREM TO

THE EXPANSION OF SERIES.

455. The method of expanding any binomial the form (ax)m, when m is any whole number