SECTION XXXV. BINOMIAL THEOREM. ART. 331. The Binomial Theorem is a general algebraical expression or formula, by which any power or root of a given quantity, consisting of two terms, is expanded into a series; the form of which, as it was first proposed by Sir Isaac Newton, being as follows: where P is the first term of a binomial, Q the second divided by m the first, the index of the power or root, and A, B, C, &c., n the terms immediately preceding those in which they are first found, including their signs or —. 332. This theorem may be applied to any particular case, by substituting the numbers or letters in the given example for P Q, m, and n, in either the above formulæ, and then finding the result according to the rule. When the index of the binomial is a whole number, the series will terminate, as observed under the article Involution; but when it is a negative or fractional number, as in the following examples, the series will proceed on ad infinitum, and will become more convergent the less the second term of a binomial is with respect to the first. EXAMPLES. 1. It is required to convert (a2+x) into an infinite series. Xx x2 3x3 a+ 3.524 3.5.7x5 &c. 2.4.6.8.10a 2a 2.4a3+ 2.4.6a2.4.6.8a7+ The pupil will readily perceive that the law of formation of the several terms of the series is sufficiently evident. And, y 3y2 3.5y3 3.5.7y+ 2.232.42 2.4.62+2.4.6.829+, &c. y 3y2 3.5y 3.5.7y* (x2-y)} 2x 2.4x3 2.4.6.x 2.4.6.8+, &c. ·=x+; + + This last equation is obtained from the former by multiplying each term of the equation by x2. 1 $=2+ 1 5 5.8 3.223.6.2++ 3.2 3.6.2 3.6.9.273.6.9.12.210+, &c. 4. What is the square root of a+b? Therefore, (a+b)1—a2+a ̄1b_a2b2 a3¿2 5a−bb1 5. What is the cube root of 7 ? 2 + &c. 8 16 128 8. It is required to convert (a—b) into an infinite series. 333. This is a general method of obtaining a series from frac tions, and other expressions, without either performing the division or extracting the root. RULE. Assume a series with unknown but constant coeffi cients of x, increasing or decreasing in the same way as if the operation was performed at length; then make this series equal to the given expression, and, clearing the equation of fractions, bring all the terms to one side, so as to make the equation = 0; next make the first term of the coefficients of the several powers of x each = 0, and there will arise as many independent equations as there are unknown coefficients, from which their values may be found and substituted for them in the assumed series. EXAMPLES. a into a series. b+x 1. Let it be required to expand b + x =A+Bx+Сx+Dx+&c.; then, multiplying both sides by b+x, and transposing a, we obtain Ab-a+ (Bb+A)x+(Cb+B)x2+(Db+C)x3+&c.=0, an equation which must be true, whatever be the value of x. Now, making the first term, and the coefficients of the several powers of ¤, each = 0, we have Ab—a=0, or A=2; Bb+a=0, or &c. And, substituting these values of A, B, C, D, &c., in the assumed series, we get 0х2 ах a b+x b b2 + &c., in which, it is obvious, that the signs are 63 alternately + and -, and the exponents, both in the numerator and denominator, increase continually by 1, that of x in the numerator being always 1 less than that of b in the denominator. |