1 1 simpler, viz. ------- and ------------------ . q^{2}_n q_n(q_n + q_{n+1}) Every simple continued fraction must converge to a definite limit; for its value lies between that of the first and second convergents and, since p_n p_{n-1} 1 p_n p_{n-1} --- ~ ------- = ------------, Lt. ----- = Lt. -------, q_n q_{n-1} q_{n}q_{n-1} q_n q_{n-1} so that its value cannot oscillate. The chief practical use of the simple continued fraction is that by means of it we can obtain rational fractions which approximate to any quantity, and we can also estimate the error of our approximation. Thus a continued fraction equivalent to [pi] (the ratio of the circumference to the diameter of a circle) is 1 1 1 1 1 1 3 + - -- -- --- -- -- 7 + 15 + 1 + 292 + 1 + 1 + ... of which the successive convergents are 3 22 333 355 103993 --, --, ---, ---, ------, &c., 1 7 106 113 33102 the fourth of which is accurate to the sixth decimal place, since the error lies between 1/q4q5 or .0000002673 and a6/q4q6 or .0000002665. Similarly the continued fraction given by Euler as equivalent to 1/2(e -1) (e being the base of Napierian logarithms), viz. 1 1 1 1 1 -- -- -- -- -- 1 + 6 + 10 + 14 + 18 + ..., may be used to approximate very rapidly to the value of e. For the application of continued fractions to the problem "To find the fraction, whose denominator does not exceed a given integer D, which shall most closely approximate (by excess or defect, as may be assigned) to a given number commensurable or incommensurable," the reader is referred to G. Chrystal's _Algebra_, where also may be found details of the application of continued fractions to such interesting and important problems as the recurrence of eclipses and the rectification of the calendar (q.v.). Lagrange used simple continued fractions to approximate to the solutions of numerical equations; thus, if an equation has a root between two integers a and a + 1, put x = a + 1/y and form the equation in y; if the equation in y has a root between b and b + 1, put y = b + 1/z, and so on. Such a method is, however, too tedious, compared with such a method as Homer's, to be of any practical value. The solution in integers of the indeterminate equation ax + by = c may be effected by means of continued fractions. If we suppose a/b to be converted into a continued fraction and p/q to be the penultimate convergent, we have aq - bp = +1 or -1, according as the number of convergents is even or odd, which we can take them to be as we please. If we take aq-bp = +1 we have a general solution in integers of ax + by = c, viz. x = cq - bt, y = at - cp; if we take aq - bp = -1, we have x = bt - cq, y = cp - at.