when x is indefinitely diminished is a.. Suppose that the series consists of an infinite number of terms. 29 Let b be the greatest of the coefficients a,, ɑ2, ɑzi let us denote the given series by a+S; then S<bx+ bx2 + bx3 + Thus when x is indefinitely diminished, S can be made as small as we please; hence the limit of the given series is a.. If the series consists of a finite number of terms, S is less than in the case we have considered, hence a fortiori the proposition is true. 269. In the series 3 by taking x small enough we may make any term as large as we please compared with the sum of all that follow it; and by taking x large enough we may make any term as large as we please compared with the sum of all that precede it. The ratio of the term ax" to the sum of all that follow When x is indefinitely small the denominator can be made as small as we please; that is, the fraction can be made as large as we please. Again, the ratio of the term a," to the sum of all that precede it is When x is indefinitely large, y is indefinitely small; hence, as in the previous case, the fraction can be made as large as we please. 270. The following particular form of the foregoing proposition is very useful. consisting of a finite number of terms in descending powers of x, by taking a small enough the last term a can be made as large as we please compared with the sum of all the terms that precede it, and by taking a large enough the first term ax" can be made as large as we please compared with the sum of all that follow it. Example 1. By taking n large enough we can make the first term of n1- 5n3 − 7n+9 as large as we please compared with the sum of all the other terms; that is, we may take the first term n1 as the equivalent of the whole expression, with an error as small as we please provided n be taken large enough. Example 2. Find the limit of 3.x3- 2x2-4 when (1) x is infinite; (2) x is zero. (1) In the numerator and denominator we may disregard all terms but 3x3 3 the first; hence the limit is ྃ་ 5x3 or " Let P denote the value of the given expression; by taking logarithms we have . Hence the limit of log P is 2, and therefore the value of the limit required is e2. If we put x=a+h, then h will approach the value zero as x approaches the value a. and when h is indefinitely small the limit of this expression There is however another way of regarding the question; for and if we now put xa the value of the expression is x2 + ax 2a2 If in the given expression we put xa before 0 simplification it will be found that it assumes the form the value of which is indeterminate; also we see that it has this form in consequence of the factor x-a appearing in both numerator and denominator. Now we cannot divide by a zero factor, but as long as x is not absolutely equal to a the factor a may be removed, and we then find that the nearer x approaches to the value a, the nearer does the value of the fraction approximate to or in accordance with the definition of х 3 2' 272. If f(x) and (x) are two functions of x, each of which becomes equal to zero for some particular value a of x, the When x=3, the expression reduces to the indeterminate form; but by removing the factor x-3 from numerator and denominator, the fraction When x=3 this reduces to which is therefore the becomes required limit. 1 4' x2-2x+1 x2+2x+1° Example 2. The fraction N - a−√x+a 3x becomes when x=a. x-a To find its limit, multiply numerator and denominator by the surd conjugate to √3x-a-√x+a; the fraction then becomes To find its limit, put x=1+h and expand by the Binomial Theorem. Thus the fraction 273. Sometimes the roots of an equation assume an indeterminate form in consequence of some relation subsisting between the coefficients of the equation. H. H. A. 15 a simple equation is indefinitely great if the coefficient of x is indefinitely small. a' If ab' - a'b= 0, then x and y are both infinite. b a b In this case = m suppose; by substituting for a, b, the second equation becomes ax + by + c' If is not equal to c, the two equations ax + by + c = 0 and m ax+by+ O differ only in their absolute terms, and being inconsistent cannot be satisfied by any finite values of x and y. Here, since be' – b'c = 0 and ca'-c'a = 0 the values of x and y 0 each assume the form and the solution is indeterminate. In fact, in the present case we have really only one equation involving two unknowns, and such an equation may be satisfied by an unlimited number of values. [Art. 138.] The reader who is acquainted with Analytical Geometry will have no difficulty in interpreting these results in connection with the geometry of the straight line. |