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Then, since (1+x) is the nth power of (1+x); and, as above shown, (1+x)=1 tax+62+cxt-dx++ &c., such a series must be assumed for (1+x)ī, that, when raised to the nth power, will give a series of the form 1 tax+bx+cx'tdx9+ &c.
But the nth or any other integral power of the series it px+qpatrz*43.24+ &c. will be found, by actual multiplication, to give a series of the form here mentioned; whence, in this case, also, it necessarily follows, that
(1+x)n =1+px+qx+trmitosx* f- &c. And if each side of this last expression be raised to the nth power, we shall have (1+x)"=[1 +(px+qx2 rx8-sz* + &c.)]”; or, by actual involution, it-metb2P+cx?+ &c. =iton(px+qx® + &c.) +- &c.
Whence, by comparing the coefficients of c, on each side of this last equation, we shall have, according to (Art. 435), , npm, or ps ; eo that, in this case,
(1+x)=1+ 2+9x® +rx3 +sr*+&c... where the coefficient of the second term, and the several powers of I, follow the same law as in the case of integral powers.
441. Lastly, if the index be negative, it will be found hy division as above, that (1+x) or the equivalent expression, 1
: 1. -::- q'x— &c. (3), (1+x) ñ 1+2x+qx&c.
& where the series still follows the same law as before.
442. And as the several cases, (1,2,3), here given, are of the same kind with those that are designed to be expressed in universal terms, by the general formula ; it is in vain, as far as regards the first two terms, and the general form of the se. ries, to look for any other origin of them than what may
be derived from these, or other similar operations.
443. Hence, because (a+a)m=am (1+ if there be assamed (a+*) "=am+man-x+Bx*+Cx'+Dx* &c.;
which will be more commodious, and equally answer the desigo proposed, (1+) =1+4, (á) +17 (5)*+,(*):+ &c. .... (4), it will only remain to determine the values of the coefficients A,,A,, A., &c. and to show the law of their dependeoce on the index (m) of the operation by which they are produced.
414. 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+(y+.z)]m=[(1+y)+z]m, which being all identical expressions, when taken according to the above form, will evidently be equal to each other.
445. Whence, as the numeral coefficients A,, A2, A2, &c. of the developed formulæ, 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+(y+z)]m=1+A, (y+2)+ A,(y+z)+ &c. [(1+y)+2]m=(1+y)m+A,:(1+y)m-i+A,z*(1+y)m-+ &c.
And, consequently, by raising the several terms of the first of these series to their proper powers, aod putting 1+y=P in the latter, we shall have 1 t-A, (y+z)+A,(y2 +2yz+z)+Az(33+34ʻz +-3yzé + zo)+ &c.=pm +Apr-12+Apm-2a+A-323+ &c.
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 : 1+A,
z+A 22+AZ 23+ &c. (5). A,y+2A y +3A3y +4A,
&c. -p? + A,pmadz+A2pm-224+Apm-323+ &c. To which equation it is evident, that both y
apd z are inde. terminate, and independent of the values of A,, Ag, Ag, &c.; since the result here obtained arises solely from the substitution of the sum of these quantities for in equation (4).
417. Hence, as the first terms and the coefficients, or multipliers of the like powers of z, in these two expressions, are, in this case, identical, (Art. 435), we shall have, by comparing the first column of the left-hand member with the first term of that on the right,
1+A,y+A2yo+Azy+A2y4+ &c. =pm. which is an identity that verifies itself ; since, by hypothesis, (1+y)"=pm, and, according to the general formula, (1+y)" =l+A,Y+Azyo+A343+ &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,
A, +2A2y+31,42 +4A.yo=A, pm-'; which will be sufficient, independeatly of the rest of the terms for determining the values of the coefficients A,, Ag, Az, &c. For since A,pm-i=A,
(1+4,y to Ayo +A343 +
P ity &c.), the equating this series with the last, and multiplying the left-hand side by ity, will give [A, +2A2y+3A2y+ + &c.](1+y)=A; + A,A,y+ A, A2yo+A, A,+-&c.
And, therefore, by actually performing the operation, and arranging the terms accordingly, we shall have A, +2A, y+3A2 ya+4A
y3 +&c. ta, +2A, +3A2
=A, + A,A,y+ A, A2y + A, A2y3 to &c. 449. From which last identity, there will obviously arise, by equating the homologous terms of its two members, the following relations of the coefficients : A=A,
A 2A,=A, A, -A
2 3A,=A, Ag-2A2
1,, 4A,=A, A,-3A,
An-:-[A, -(-1); nAn=A, An-,-(0–1)An-1
And, consequently, as the coefficient A, of the second term of the expanded binomial, has been sbown to be equal, in all
And if !
cases, to the index (m) of the proposed binomial, the last of these expressions will become of the form
2.3.4 mm-1).(-2).(m-3) Ana
i 126.96.36.199 where the law of the continuation of the terms, from A4 to the general term An, 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
m(m--1) m(m-- 1)(m--2) +mam-+
am-3,3 &c. 2
2.3. be substituted in the place of to the same formula will, in that case, be expressed as follows:
3+ &c.] ;
am-3.23 &c. 2.4 Where it is to be observed, that the series, in each of these cases, will terminale at the (mt1)th term, when m is a whole positive pumber ; but if m be fractional or negative, it will proceed ad infinitum ; as veither the factors
-3, &c. can then become =0.
**(1-5) =a"[1- ) + 2" (*)
451. To this we may add, that in the two last instancos 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 in these formulæ, are each equal to 1, we shall have, agreeably to such a substitution, (a+x)"=(1+1)=2"=1+m+ 72(mn 1), mm-1). (mm2), n(m-1). (
(A-2)"=(1-1)=0m=0=1-mt mm---1)_mm-1). (m-2) 4(m
in(m - 1)m -- 2). (m-3) 2.3
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 mth 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+*) =a +ox a 0(0-1) 0-22
. 2 where it is evident that the series terminates at the first term (); since the coefficient of every successive term involves Ò for one of its factors ; therefore (a+x)=a'=1, (Art. 86). And, if a=c; then (a--X°=a=1, that is, o=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 oo appears to admit of an infinity of numerical values; because it is equal to f, which is the mark of indetermination ; yet it is plain, from what is above shown, that O' is only one of the values of $, which, in that particular case (Art. 167), where om
50°= is equal to unity. The intelligent reader is reOm ferred to BONNYCASTLE's Algebra, 8vo. vol. ii. Also, LAGRANGE's Theorie des Fonctions Analytiques, and Leçons sur le Lalcul des Fonctions,