Imágenes de páginas
PDF
EPUB

Assume

a + bx
a' + b'x + c'x2

= P+Qx + Rx2 + Sx3 + &c.

Clearing of fractions, and transposing, we get

Pa' + Qa' | x+ Ra' | x2 + Sa' | x3 + &c. = 0.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][subsumed][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][subsumed][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

from which we see that, commencing at the third, each co-efficient is formed by multiplying the two which precede it, re

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

From this law of formation of the co-efficients, it follows that the third term, and every succeeding one, is formed by multi

plying the one that next precedes it by

[ocr errors]

b'

x, and the second

ad,

preceding one by ——2, and then taking the algebraic sum of

these products: hence,

a'

[blocks in formation]

This scale contains two terms, and the series is called a recurring series of the second order. In general, the order of a recurring series is denoted by the number of terms in the scale of the series.

The development of the fraction

a + bx + cx2

a' + bx + c2x2 + d'x3'

gives rise to a recurring series of the third order, the scale of which is,

[merged small][merged small][ocr errors][subsumed][merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors]

gives a recurring series of the nth order, the scale of which is

[merged small][merged small][merged small][ocr errors][merged small]

202. It has been shown (Art. 60), that any expression of the form mym, is exactly divisible by zy, when m is a positive whole number, giving,

[ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

The number of terms in the quotient is equal to m, and if we suppose z = y, each term will become m-1; hence,

[blocks in formation]

The notation employed in the first member, simply indicates what the quantity within the parenthesis becomes when we make

y = 2.

We now propose to show that this form is true when m is fractional and when it is negative.

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

If now, we suppose y = z, we have v = u, and since are positive whole numbers, we have

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors]

p

[merged small][ocr errors][merged small]

and

Second, suppose m negative, and either entire or fractional.

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

If, now, we make the supposition that yz, the first factor of the second member reduces to

2m, and the second factor, from the principles just demonstrated, reduces to mm;

hence,

[merged small][merged small][ocr errors][merged small][merged small]

We conclude, therefore, that the form is general.

203. By the aid of the principles demonstrated in the last article, we are able to deduce a formula for the develop ment of

(x + a)m,

when the exponent m is positive or negative, entire or

Let us assume the equation,

fractional

[merged small][merged small][ocr errors][merged small]

in which, P, Q, R, &c., are independent of z, and depend upon 1 and m for their values. It is required to find such values for them as will make the assumed development true for every possible value of z.

If, in equation (1) we make z = 0, we have

P=1.

[ocr errors]

Substituting this value for P, equation (1) becomes,

[ocr errors]

(2).

(1 + z)m = 1 + Qz + Rz2 + Sz3 + &c. Equation (2) being true for all values of z, let us make z = y;

whence,

(1+y)=1+Qy+ Ry2+ Sy3 + &c.

(3).

Subtracting equation (3) from (2), member from member, and dividing the first member by (1 + 2) − (1 + y), and the second member by its equal zy, we have,

[blocks in formation]

If, now, we make

member of equation

[blocks in formation]

1+z =

[subsumed][ocr errors][ocr errors][merged small][merged small][merged small]

1+y, whence zy, the first

(4), from previous principles, becomes m (1+2)-1, and the quotients in the second member become respectively,

[blocks in formation]

Substituting these results in equation (4) we have,

m (1+2)-1=Q+2Rz+3Sz2+4T23+ &c.

[ocr errors]
[ocr errors][merged small]

Multiplying both members of equation (5) by (1 + 2), we find, m(1 + z)m = Q+2Rz+3S | 22 +47 | 23+ &c. + Q + 2R +3S

[ocr errors]

(6).

we have

If we multiply both members of equation (2) by m, m (1 + z)m = m + mQz + mRz2 + mSz3 + mT21 + &c. (7).

[ocr errors]

The second members of equations (6) and (7) are equal to each other, since the first members are the same; hence, we have the equation,

m+mQz+mRz2+mSz3+&c.=Q+2Rz+3S22+47 23+ &c + Q+2R +3S[

(8)

This equation being identical, we have, (Art. 195),

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

Substituting these values in equation (2), we obtain

[merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

3)

4

2)

z3

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

Hence, we conclude, since this formula is identical with that deduced in Art. 136, that the form of the development of (x+a)" will be the same, whether m is positive or negative, entire or fractional.

It is plain that the number of terms of the development, when m is either fractional or negative, will be infinite.

[merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
« AnteriorContinuar »