terminates, may have a determinate value; and it should likewise be observed, that from this branch of mathematics have been derived inventions of the utmost importance; for which reason the subject deserves to be studied with every possible 1+x into an infinite series. ANS. b2 × (1−x + x2 — x3 + x* — x5 + &c.) ad inf. ANS. 1 264 -&c. ad inf. α 5. Resolve 16+ into an infinite series. 1-b+b2 ANS. a × (1+b− b 3 − b + + b ® + b7-b9+ &c.) ad inf. into an infinite series. ANS. x2 -x+x6 — x 8 + x 10 — x 11⁄2 + &c. ad inf. ° 12+ 7. Resolve 1 1+2y+3x into an infinite series. ANS. 1−(2y+3x) + (2y +3x)2 — (2y +3x)3 +(2y+3x)+ - &c. ad inf. CHAP. XXXV. ON THE EXPANSION OF IRRATIONAL POWERS IN INFINITE SERIES. IN Chap. XXXII. we have shown the method of finding any power of the root a+b, however great the exponent of the pover may be, and that by representing any such exponent by n, the general expression for the expansion of the power will be, a+b)" =a" + na"-18 + n. (n-1) 2 1. n. (n−1.)(n-2) an-3b3+ &c.; a formula which equally serves for the powers of a-b, since it would then be only necessary to change the signs of the even terms. be 286. These formulas are remarkably useful, since they serve likewise to express all kinds of radicals. We have already seen that all surds, or irrational quantities, may represented in the forms of powers, by fractional indices or exponents, and that ✔a=a*; a=a*, and /a=a*, &c. Inlike manner, therefore, √/a+b=a+b)*; 3⁄4 / a+b=a+b) 3 and a+b=a+b+, &c. 287. In order therefore to find the square root of a +b we have only to substitute for the exponent n the fraction in the above general formula, and we shall have first for the co 1 3 5 2 3 n &c. Or we might express these powers of a in another form, as, This being the case therefore, the root of a+b will be expressed in the following manner : 288. Now if a be a square number, we may assign the value of a, and consequently the square root of a+b may in that case be represented in an infinite series, without any radical sign. Assume, for example, ac2, then a=c, and From hence it appears that there is no number of which we may not extract the square root in the same manner, since every number may be divided into two parts, of which one may be a square number represented by c2. For example, if it were required to extract the square root of 6, we might represent 6=4+2, where c2=4 and b=2, and consequently the square root of 6 will be, Now if we take only the two first terms of this series, we 1 5 shall have 2+2=2, of which the square is 1 4 greater than 6 by; but if we take three terms, we shall 7 39 1521 256 have 216=16, the square of which is too little by 5 Since, however, in this example approaches very nearly 2 to the true value of 6, we will take for 6 the equivalent quantity 400' 20' C and b 400' and taking again the two leading terms of the series, we shall have, 289. We may in the same manner express the cube root 1 of a+b by an infinite series. For in this case n= and ; and with regard to the 1 3 พ- 1 n-1 &c. Therefore 3⁄4a+b, or a + b2 Ja 5 $/a i an-2 a $/a as 243 b. + a2 81 If a therefore be a cube, such that ac, we shall then have /ac, and the radical signs will vanish, for we shall have, 290. Thus then we have arrived at a formula, by means of which we are enabled to find by approximation the cube root of any number, since any number may in this case. also be divided into two parts, as c3+b, of which the first is a cube. In order, for instance, to determine the cube root of 2, we shall represent it by its equivalent 1+1, so that c=1, By pursuing the same course we shall approach nearer tothe root, and the more rapidly in proportion as we take a greater number of terms; and the same theorem will apply. to the extraction of any root whatever. |