So also in succession But = Pn-19n-2-Pn-29n-1 (Pn-2In-3-Pn-In-2), = P3L2 - P2I3 = − (P2Y1 — P1J2). P2- P12 = (a1a, + 1) − a,a,= 1. Hence Pln-- Pn-19n = (− 1)”.......... .(i). .(ii). PnIn-1-Pn-1In = (− 1)"1 and Pn - Pn-1 (− 1)"-1 In In-1 II. Every common measure of P and = In InIn-1 must also be a measure of PnIn-1 - Pn-19, that is, from I., a measure of +1. Hence Pn and I can have no common measure. Thus all convergents are in their lowest terms. Hence F-P2 _ Pn +λPn-1 _ Pn _ λ (Pn-1In — PnIn-1) In = In + λIn-1 In (-1)"-1X In (In + λIn-1)* = In In +λn-1) In-13 Now is less than 1, and In is greater than -1; hence FP is less than F~P-1 Thus any convergent is nearer to the continued fraction than the immediately preceding convergent, and therefore nearer than any preceding convergent. IV. If any fraction, suppose, be nearer to a continued y fraction than the nth convergent, then must from III. be y also nearer than the (n-1)th convergent; and, as the continued fraction itself lies between the nth and the (n - 1)th convergents [Art. 350], it follows that must also lie between these convergents. y .. y must be > In (Pn-1Y ~ In-1). Hence, as all the quantities are integral, y must be greater than qn' Thus every fraction which is nearer to a continued fraction than any particular convergent must have a greater denominator than that convergent. V. We have seen in III. that where X is a positive quantity less than unity. Thus any convergent to a continued fraction differs from the fraction itself by a quantity which lies between 1 and 1 did2 d1(d1 + d2)' where d, and d, are respectively the denominators of the convergent in question and the next succeeding convergent. Ex. 1. Shew that, if pr/qr be the rth convergent to the continued Ex. 3. Shew that, if pr/q, be the rth convergent of then will Pn+1=a¶n• Assume that the theorem is true for all values up to n; then Pn+1=bpn+apn-1=ba¶n-1+a2¶n-2 =a (bqn-1+aqn-2) = a¶n• Hence, if the theorem be true for any two consecutive values of n, it will be true for the next greater value. Also it is easily seen to be true when n=1 and when n = 2: it must therefore be true for all values of n. EXAMPLES XXXVI. 1. Shew that, if P1, P2, P3 be three successive convergents 1 2 3 to any continued fraction with unit numerators, then will P3-P13-I1 = P2 · L' 2 3. Two graduated rulers have their zero points coincident, and the 100th graduation of one coincides exactly with the 63rd of the other: shew that the 27th and the 17th more nearly coincide than any other two graduations. 4. Shew that, if a,, a,,......a be in harmonical progression ; 1 1 then will 5. ༦. = a 2-2 n-1 Shew that 1 2 10. If p1 be the numerator of the nth convergent to the b1 + b2 + b2 + 3 every successive four of the series p1, P2, P3,...; and find what the relation is. |