241. Let us apply the above principles in developing into a series the expression α a + b'x' It is plain, that any expression equal to the above, must contain x, and the quantities a, a', b'. Let us then assume a = A + Bx + Сx2 + Dx3 + Exa + &c. (1), . a + b'x in which the co-efficients A, B, C, D, &c., are functions of a, a', b, and independent of x. These are called, indeterminate co-efficients. It is required to find their values in terms of a, a', b', on which they depend. For this purpose, multiply both members of equation (1) by a+b'x. Arranging the result with reference to the powers of x, and transposing a, we have 0 = { S Aa + Ba x + Ca' | x2 + Da' | x3 + Ea′ | x++ + + Db' and since this equation is satisfied for any value of x, we have It is plain that the terms will be alternately plus and minus, and that the co-efficient of any term is formed by multiplying that 242. The method of indeterminate co-efficients requires that we should know the form of the development. The terms of the development are generally arranged according to the ascending powers of x, commencing with the power ao; sometimes, however, this form is not applicable, in which case, the calculus detects the error in the supposition. For example, develop the expression Let us suppose that 1 3x 22 = A + B + Сx2 + Dx3 + whence, by reducing to entire terms, and arranging with reference Now, the first equation, 10, is absurd, and indicates that the above form will not develop the expression 1 - But if 3x x2 and make x 3 -X = A + Bx + Сx2 + Dx3 + we shall have, after the reductions are made, 0 = which gives the equations 3A 10, 3B B + 3D | 23 + C = D = 81 that is, the development contains a term with a negative expo nent. Recurring Series. 243. The development of algebraic fractions by the method of indeterminate co-efficients, gives rise to certain series, called recurring series. A recurring series is the development of a rational fraction involving x, made according to a fixed law, and containing the ascending powers of x in its different terms. It has been shown in Art. 241, that the expression in which each term is formed by multiplying that which precedes This property of determining one term of the development from those which precede, is not peculiar to the proposed fraction; it belongs to all rational algebraic fractions, and may be thus expressed; viz., Every rational fraction involving x, may be developed into a series of terms, each of which is equal to the algebraic sum of the products which arise from multiplying certain terms of a particular expression, by certain of the preceding terms of the series. The particular expression, from which any term of the series may be found, when the preceding terms are known, is called the scale of the series; and that from which the co-efficient may be formed, the scale of the co-efficients. In the preceding series, the scale is b' called a recurring series of the first order; and of the co-efficients. Let it be required to develop Assume a + bx a + bx + c2x2 x, and the series is = A + Bx + Сx2 + Dx3 + Ex1 + reducing to entire terms, and transposing, we have Bax + Ca | x2 + Da' | x3 + Ea' | x2 + Aa From which we see, that the first two co-efficients are not obtained by any law; but commencing at the third, each co-efficient is formed by multiplying the two which precede it, respectively, required co-efficient by viz., that which immediately precedes the b a" that which precedes it two terms by ——, and taking the algebraic sum of the products. Hence, a'' From this law of the formation of the co-efficients, it follows that the third term of the series, Cx2, is equal to Hence, each term of the required series, commencing at the third, is obtained by multiplying the two terms which precede, and taking the sum of the products: hence, this last expression is the scale of the series. Recurring series are divided into orders, and the order is estimated by the number of terms in the scale which involve x. The series obtained in the preceding Art. is of the second order. In general, an expression of the form gives a recurring series of the nth order, the scale of which is REMARK. It is here supposed that the degree of x in the numerator is less than it is in the denominator. If it was not, it would first be necessary to perform the division, arranging the quantities with reference to x, which would give an entire quotient, plus a fraction similar to the above. Performing the division, we find the quotient to be x-7, |