Elements of Algebra

296 ELEMENTS SECT. IV. 799. But if the beginning of the series exceed the root, we shall not approximate towards that root at all; for when the equation has more than one root, the series gives by approximation only the greatest: and we do not find one of the less roots, unless the first terms have been properly chosen for that purpose. This will be illustrated by the following example. Let there be given the equation $x^2 = 4x - 3$, whose two roots are $x = 1$, and $x = 3$. The formula for the series is $r = 4q - 3p$, and if we take 1, 1, for the first two terms of the series, which consequently expresses the least root, we have for the whole series, 1, 1, 1, 1, 1, 1, 1, 1, &c. but assuming for the leading terms the numbers 1, 3, which contain the greatest root, we have the series, 1, 3, 9, 27, 81, 243, 729, &c. in which all the terms express precisely the root 3. Lastly, if we assume any other beginning, provided it be such that the least term is not comprised in it, the series will continually approximate towards the greatest root 3; which may be seen by the following series: Beginning, 0, 1, 4, 13, 40, 121, 364, &c. 1, 2, 5, 14, 41, 122, 365, &c. 2, 3, 6, 15, 42, 123, 366, 1095, &c. 2, 1, -2, -11, -38, -118, -362, -1091, -3278, &c. in which the quotients of the division of the last terms by the preceding always approximate towards the greater root 3, and never towards the less. 800. We may even apply this method to equations which go on to infinity. The following will furnish an example: $x^\infty = x^{\infty-1} + x^{\infty-2} + x^{\infty-3} + x^{\infty-4} +$, &c. The series for this equation must be such, that each term may be equal to the sum of all the preceding; that is, we must have 1, 1, 2, 4, 8, 16, 32, 64, 128, &c. whence we see that the greater root of the given equation is exactly $x=2$; and this may be shewn in the following manner. If we divide the equation by $x^\infty$, we shall have $1 = \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + \frac{1}{x^4} +$, &c. a geometrical progression, whose sum is found $= \frac{1}{x-1}$; so