Assume a + bx = P+Qx + Rx2 + Sx3 + &c. Clearing of fractions, and transposing, we get Pa' + Qa' | x+ Ra' | x2 + Sa' | x3 + &c. = 0. from which we see that, commencing at the third, each co-efficient is formed by multiplying the two which precede it, re 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 b' x, and the second ad, preceding one by ——2, and then taking the algebraic sum of these products: hence, a' 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, gives a recurring series of the nth order, the scale of which is 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, The number of terms in the quotient is equal to m, and if we suppose z = y, each term will become m-1; hence, 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. If now, we suppose y = z, we have v = u, and since are positive whole numbers, we have p and Second, suppose m negative, and either entire or fractional. 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, 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 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. Substituting this value for P, equation (1) becomes, (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, If, now, we make member of equation 1+z = 1+y, whence zy, the first (4), from previous principles, becomes m (1+2)-1, and the quotients in the second member become respectively, Substituting these results in equation (4) we have, m (1+2)-1=Q+2Rz+3Sz2+4T23+ &c. 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 (6). we have If we multiply both members of equation (2) by m, m (1 + z)m = m + mQz + mRz2 + mSz3 + mT21 + &c. (7). 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), Substituting these values in equation (2), we obtain 3) 4 2) z3 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. |