UNDETERMINED COEFFICIENTS 588. Coefficients assumed in the demonstration of a principle or the solution of a problem, whose values, not known at the outset, are to be determined by subsequent processes, are called Undetermined Coefficients. put To expand (x 1)(x + 1)(x − 2) without actual multiplication, (x − 1)(x + 1)(x − 2) = x2 + Ax2 + Bx + C, (1) and determine A, B, and C by means of the fact that (1) is an identity and must be true for all values of x. By solving these three conditional equations, the coefficients A, B, and C, undetermined in (1), are found to be A2, B = − 1, C=2. .. (x − 1)(x + 1)(x − 2) = x3 − 2 x2 − x + 2. 589. It has been shown that the infinite geometrical series 1 + x + x2 + 203 +.... is convergent when a is numerically less than 1, and that the sum Then, § 420, lim. (1+x+x2+x3+...)x=0 = lim. .. lim.(x+x+2°3 + ···)x÷0 = 1−1 = 0. Hence, if N is any constant number, § 422, lim. [N(x + x2 + æ3 + ···)]x÷0 = N × 0 = 0. (1) This result is useful in finding the limit of the infinite series A+ Bx + Сx2 + Da3 +..., when x=0. For let N be positive, and numerically equal to the greatest of the coefficients, B, C, D, .... ... Then, Bx+Cx2+ Dæ3 + ··· < N (x + x2 + æoa3 + ···), numerically. Therefore, by (1), lim. (Bx+Cx2 + Dæ3 + ···) x=0 = 0. Therefore, § 421, Hence, PRINCIPLE. When x approaches the limit zero, the sum of the lim. (A + Bx + Сx2 + Dx23 + ···•)x±0 = A. first n terms of the series A+ Bx + Сx2 + Dx3 + A as a limit as n increases without limit. (1) 590. Let A+ Bx + Сx2 + ··· = A' + B'x + C'x2 + ···, for every value of x that makes both series convergent. Then, for such values of x, § 568, the sum of the first n terms of each series approaches a limit as noc, and by § 419 these limits are equal; that is, = lim. (A+B+Cx2+...) lim. (A' + B'x + C''x2 + ···), for every value of x that makes both series convergent. Since, § 589, lim. (A + B + Cx2+) x=0 = A, and lim. (A'B' + C'x2+) x0 = A', by § 568 both series are convergent when x 0. .. lim. (A + Bx +Сx2+ ···)x÷0 = lim. (A' + B'x + C'x2 + ···)x=0; Since (2) (3) 0 but is not equal to zero, the members of (3) may be divided by x. ... .. B+CxDx2 + ··· = B' + C'x + D'x2 + ···, for all values of x that make the given series convergent. From (4), by the reasoning applied to (1), = (4) lim. (B+C+Dx2+) x=0 lim. (B' + C'x + D'x2 + ...)x=0; that is, B B', whence a (C + Dx + ···) = x (C" + D'x + ...). Similarly, CC, D= D', etc. Hence, = PRINCIPLE OF UNDETERMINED COEFFICIENTS. If two series arranged according to the ascending powers of x are equal for every value of x that makes both series convergent, the coefficients of the like powers of x are equal each to each. Since lim. (A + Bx + Câ2 + ···)=0= A also when the series is finite (§ 423), the Principle of Undetermined Coefficients applies when one or both of the series are finite. 591. An algebraic expression is said to be developed when it is transformed into a series whose sum, if it is finite, or the limit of whose sum, if it is infinite, is equal to the given expression. DEVELOPMENT OF FRACTIONS EXAMPLES 592. 1. Develop the fraction 1+2x SOLUTION The first term of the development is evidently 11, or 1; and since the denominator is not exactly contained in the numerator, the development is an infinite series beginning with 1 and proceeding according to ascending powers of x. To determine the coefficients of the various powers, assume true for all values of x that make the second member a convergent series. Clearing of fractions and collecting terms, The first member may be regarded as the infinite series, which has a definite sum for all values of x, while the second member is an infinite series having a definite sum for such values of x as make the series assumed in (1) convergent. Therefore, since (2) is true for all values of x that make the assumed series convergent, by the principle of undetermined coefficients the coefficients of like powers of x in (2) may be equated. 2-x+2x2= A + B x + Cx2 + D c3 + Ex + Equating the coefficients of like powers of x, § 590, and solving, DEVELOPMENT OF SURDS EXAMPLES 593. 1. Develop the expression Va+a by the use of undetermined coefficients. SOLUTION Assume √a + x = A + Bx + Сx2 + Dx3 + Ex2 + · Squaring, a + x = A2 + 2 ABx + B2 | x2 + 2 AD x3 + C2x2 + ··.. +2 BC+2 AE +2 АС Equating the coefficients of like powers of x, § 590, +2 BD 5√a 128 at 5x4 128 at The given surd may also be developed by the extraction of the root indicated or by the use of the binomial formula. But whatever the method of development, the series obtained is equal to the surd only for such values of x as make the series convergent. |