XLVIII. PARTIAL FRACTIONS AND INDETERMINATE COEFFICIENTS. 642. An algebraical fraction may be sometimes decomposed into the sum of two or more simpler fractions; for example, The general theory of the decomposition of a fraction into simpler fractions, called partial fractions, is given in treatises on the Theory of Equations and on the Integral Calculus. (See Theory of Equations, Chap. XXIV., Integral Calculus, Chap. II.) We shall here only consider a simple case. ax2 + bx + c be a fraction, the denominator (x-a) (x-5) (x − y) 643. Let of which is composed of three different factors of the first degree with respect to x, and the numerator is of a degree not higher than the second with respect to x; this fraction can be decomposed into three simple fractions, which have for their denominators respectively the factors of the denominator of the proposed fraction, and for their numerators certain quantities independent of x. To prove this, assume where A, B, C are at present undetermined; we have then to shew that such constant values can be found for A, B and C, as will make the above equation an identity, that is, true whatever may be the value of x. Multiply by (x − a) (x – ẞ) (x − y); then all that we require is that the following shall be an identity, ax2 + bx + c = A (x − ẞ) (x − y) + B (x − a) (x − y) + C (x − a) (x − ß); this will be secured if we arrange the terms on the right hand according to powers of x, and equate the coefficient of each power to the corresponding coefficient on the left hand; we shall thus obtain three simple equations for determining A, B and C. PARTIAL FRACTIONS AND INDETERMINATE COEFFICIENTS. 391 644. The method of the preceding Article may be applied to any fraction, the denominator of which is the product of different simple factors, and the numerator of lower dimensions than the denominator. The preceding Article however is not quite satisfactory, because we do not shew that the final equations which we obtain are independent and consistent. But as we shall only have to apply the method to simple examples, where the results may be easily verified, we shall not devote any more space to the subject, but refer the student to the Theory of Equations and the Integral Calculus. 2x-3 645. Suppose we have to develop x2 - 3x+2 in a series proceeding according to ascending powers of x; there are various methods which may be adopted. We may proceed by ordinary algebraical division, writing the divisor in the order 2-3x+x2 and the dividend in the order - 3+ 2x. Or we may develop 1 by writing it in the form (x2 - 3x + 2) ̄1, and finding x2 - 3x + 2 the coefficients of the successive powers of x by the multinomial theorem; we must then multiply the result by 2x-3. It is however more convenient to decompose the fraction into partial fractions and then to develop each of these. Thus 646. Without actually developing such an expression as the above, we may shew that the successive coefficients will be connected by a certain relation; before we can shew this it will be necessary to establish a general property of series. is always equal to zero whatever may be the value of x, the coefficients a,, α, α, must each separately be equal to zero. For since the series is to be zero whatever may be the value of x, we may put x=0; thus the series reduces to a。, which must therefore itself be zero. Hence removing this term we have always zero; divide by x, then + + ....... is always zero. Hence, as before, we infer that a, = 0. Proceeding in this way, the theorem is established. are always equal whatever may be the value of x, then a-4+ (a,-A) x + (α, − ▲2) x2 + ...... Ο is always zero whatever may be the value of x; hence we infer that ; a-A=0, a12- A1 = 0, that is, the coefficients of like powers of x in the two series are equal. The theorem here given is sometimes quoted as the Principle of Indeterminate Coefficients; we assumed its truth in Art. 542. With respect to the difficulties of the demonstration of the Principle, the advanced student may consult the Chapter on this subject in De Morgan's Algebra. If n be greater than 1, the coefficient of x" on the right-hand side is u-pu-1- qu-2; hence since there is no power of x higher than the first on the left-hand side, we must have by Art. 647, for every value of n greater than 1, And by comparing the first and second terms on each side, we have the last two equations determine u, and u,, and then the previous equation will determine u, u,, u,...... by making successively n = 2, 3, 4, ... EXAMPLES OF PARTIAL FRACTIONS AND INDETERMINATE COEFFICIENTS. Expand each of the following seven expressions in ascending powers of x, and give the general term: Expand each of the following five expressions in ascending powers of x as far as five terms, and write down the relation which connects the coefficients of consecutive terms: 13. Sum the following series to n terms by separating each term into partial fractions: х + ax + a3x (1 + x) (1 + ax) (1 + ax) (1 + a3x) ̄ (1 + a3x) (1 + a3x) 14. Sum in a similar manner the following series to n terms: + ax (1 - a2x) x (1 - ax) (1 + x) (1 + ax) (1 + a2x) * (1 + ax) (1 + a2x) (1 + a3x) + 15. Determine a, b, c, d, e, so that the nth term in the a + bx + cx2 + dx3 + ex1 expansion of 16. Shew how to decompose may be n1x"-1 ̧ tial fractions, supposing that n is the number of factors in the denominator, and that p is an integer less than n. If p be less than n-1, shew that 649. A series is called a recurring series, when from and after some fixed term each term is equal to the sum of a fixed number of the preceding terms multiplied respectively by certain constants. By constants here we mean quantities which remain unchanged whatever term of the series we consider. ...... each 650. A geometrical progression is a simple example of a recurring series; for in the series a+ar + ar3 + ar3 + term after the first is r times the preceding term. denote respectively the (n-1)th term and the nth term, then If un-1 and u |