In a similar way it can be shown that, when a is numerically less than b, n la |br-2a2 + .... (4) (a + b)" = b" + +(1)α 'a + (1) Ma2 Notice that when n is a fraction or negative, formula (3) or (4) must be used according as a is numerically greater or less than b. 8. Ex. Expand 1 to four terms. If we assume a > 4 b2, we have, by (3), Art. 7, If a < 4 b2, we should have used (4), Art. 7. Any particular term can be written as in Ch. XXVIII., Art. 6. 9. Extraction of Roots of Numbers. - Ex. Find √17 to four decimal places. We have Therefore √17 = 4.1231, to four decimal places. + 8. (x2 + y) ̃3, 4. (x — y2)−, 7. (3+2x). 8. (5 a2 – 3 b3) ̃3· 17. 5th term of (x3-x-ly). 18. 8th term of (a3/b2ba). 19. k-5th term of (1+xy)-2. 20. 2 kth term of [x2 — √(xy)]3. 21. Find the term in (3 x3 - x2y) containing x2. 22. Find the term in -1 (a +21) containing a-1. Find to four places of decimals the values of 23. √5. 24. √27. 25. /35. 26. 700. 27. 258. CHAPTER XXXV. UNDETERMINED COEFFICIENTS. § 1. METHOD OF UNDETERMINED COEFFICIENTS. 1. Upon the following principles is based an important method of changing an algebraical expression from one form to another. 2. If an infinite series a+ A1x + ɑ¿x2 + AzÃ3 +··· be convergent for values of x greater than 0, the sum of the series approaches a, as x approaches 0. Evidently, if the given series be convergent, that is, if ao + xS1 be finite, then S is finite. Therefore, by Ch. XXXII., § 2, Art. 10, xS1 = 0, when x = 0. 3. If two integral series, arranged to ascending powers of x, be equal for all values of x which make them both convergent, the coefficients of like powers of x are equal. Let ao + a1x + Azx2 + ... = bọ + b1x + b2x2 + for all values of x which make the two series convergent. Then the sums of the two series approach equal limits when x = 0. But, by the preceding article, the sum of the one series approaches ao, that of the other bo; consequently ao = bo, Since by Ch. XXXIII., Art. 21, these two series are convergent for all values of x for which the original series are convergent, they are equal for values of x other than zero, and the last equation may be divided by z. In like manner, we can prove a2 = b2, as = bз, etc. 4. The principle of Art. 3 holds with greater reason if either or both of the series be finite. The series must be equal for all values of x, if they be both finite; or, if one be infinite, for all values of x which make that series convergent. 5. The condition that the roots of the equation ax2 + bx + c = 0 are equal, given in Ch. XXI., Art. 17 (ii.), can be obtained also by applying the principle of Art. 3. If the two roots be equal, ax2 + bx + c is the square of a binomial. We therefore assume ax2 + bx + c = (Ax + B)2 = A2x2 + 2 ABx + B2. = = By Art. 3, 4a (1), 2 AB b (2), B = c (3). From (1) and (3), A=√a, B=√c. Whence, by (2), 2 √(ac)= b, or b2 = 4 ac. EXERCISES I. Assuming Ar2+ Bx + C to be the quotient in each of the following divisions, find the values of A, B, C, and the value of m so that the division will be exact: 2. [(m +5) x3 + 3 x2 + (m − 5) x + 3]+(x+3). (m + 2) x + m] ÷ (x2 - 4 x + 2). § 2. EXPANSION OF FUNCTIONS IN INFINITE SERIES. 1. We shall now give a method of expanding certain functions in infinite series by the principle of § 1, Art. 3. Rational Fractions. 2. In assuming as the expansion of a rational fraction an infinite series of ascending powers of x, we first determine with what power the series should commence. This is done by division, when both numerator and denominator are arranged to ascending powers of x. In fact, this step also determines completely the first term of the series. in a series, to ascending powers of x. Since the first term of the expansion is evidently 2, assume 2 x 1 + x wherein B, C, D, ... are constants to be determined. Clearing the equation of fractions, we obtain 2-x=2+ B | x + С x2 + D│x2 + E x2 + ···· we The series on the right is infinite; that on the left may be regarded as an infinite series with zero coefficients of all powers of x higher than the first. By § 1, Art. 3, we have Hence, substituting these values of B, C, D, ... in the assumed series, we have 2-x =2-3x+5x2 - 8 x3 + 13 x1 + ·· |