4, &c., in succession, that the first member of the formula, will, in succession, represent each term of the series; while un second member will become, for n der the same supposition, the terms of the series, If now, we suppose n∞, the value of the sum of the series will become equal to 1. 2. Required the sum of n terms of the series 1 1 + &c., to infinity. q = 1, p=2, To adapt the formula to this series, we make q and n = 1, 3, 5, 7, &c.; we then have, for the sum of ʼn terms. If, now, we suppose n = ∞, the value of the series becomes equal to one half. 3. Required the sum of n terms of the series 1 + +&c., to infinity. 2.5 3. 6 4.7 1, n = 1, 2, 3, 4, &c.: hence, 1 1 + + -13 + (+ - ( = + n + 1 + n 1 + + n + 2 when n∞. If the number of terms used is even, the upper sign will apply, the quantity within the parenthesis will become +1, and the sum of the n terms before dividing by p, is If n is odd, the lower sign is used, and the quantity within the parenthesis reduces to zero, and we have Then, since p = 2, the sum of the series when n = ∞, is 1 12 4 + + &c., to infinity. 17.21 Ans 1 CHAPTER IX. CONTINUED FRACTIONS, EXPONENTIAL QUANTITIES, LOGARITHMS, AND FORMULAS FOR INTEREST. in which a, b, c, d, &c., are positive whole numbers, is called a continued fraction. Hence, a continued fraction has for its numerator the unit 1, and for its denominator a whole number, plus a fraction which has 1 for its numerator and for its denominator a whole number plus a fraction, and so on. 249. The resolution of an equation of the form in which y> 1, and the proposed equation becomes, after changing the members, It is plain, that the value of y lies between 1 and 2. Suppose and this value will satisfy the proposed equation. For, 8x= 83 = 3/85 = 3√(23)5 = 3√(25)3 = = 25 = 32. 250. If we apply a similar process to the equation Since 200 is not an exact power, x cannot be expressed either by a whole number or a fraction: hence, the value of x will be incommensurable, and the continued fraction will not terminate, but 251. Common fractions may also be placed under the form of continued fractions. Let us take, for example, the fraction. 65 149' and divide both its terms by the numerator 65, the value of the fraction will not be 2+ tion. But this value would be too large, since the denominator used was too small. If, on the contrary, instead of neglecting the part 1 to replace it by 1, the approximate value would be which 3' would be too small, since the denominator 3 is too large. Hence, 1 therefore the value of the fraction is comprised between and 3 2 If we wish a nearer approximation, it is only necessary to op 19 erate on the fraction as we did on the given fraction 65 65 149' ber which ought to be added to 2; hence, 1 divided by 2+ 3 will be less than the true value of the fraction; that is, if we stop at the first reduction and omit the fractional numbers, the result will be too great; if at the second, it will be too small, &c. Hence, generally, if we stop at an odd reduction, and neglect the fractional part, the result will be too great; but if we stop at an even reduction, and neglect the fractional part, the result will be too small. |