Continued fractions are generally written for convenience in the form 349. The fraction obtained by stopping at any stage is called a convergent of the continued fraction. Thus a and ac + b C are respectively the first and second, third, &c. elements of the continued fraction. 350. In a continued fraction of the form a + b d c+e+ where a, b, c, &c. are all positive, the convergents are alternately less and greater than the fraction itself. b c+ For the first convergent is too small because the part b ... is omitted; the second convergent, a + is too - с great because the denominator is really greater than c; d then again, the third is too small, because c + is greater e 351. In order to find any convergent to a continued fraction, the most natural method is to begin at the bottom, as in Arithmetic: thus If only one convergent has to be found, this method answers the purpose; but there would be a great waste of labour in so finding a succession of convergents, for in finding any one convergent no use could be made of the previous results: the successive convergents to a continued fraction are, however, connected by a simple law which we proceed to prove. 352. To prove the law of formation of the successive convergents to the continued fraction Now the third convergent can be written in the form from which it appears that its numerator is the sum of the numerators of the two preceding convergents multiplied respectively by the denominator and numerator of the last element which is taken into account; and a similar law holds for the denominator. We will now shew by induction that all the convergents after the second are formed according to the above law provided there is no cancelling at any stage. For, assume that the law holds up to the nth convergent, for which the last element is a-1/-1, and let Pr/q, denote the rth convergent; then by supposition Pn=bn-1Pn-1+αn-1Pn-2 and In= bn-19n-1 + ɑn-19n-2· · ·(i). Then the (n+1)th convergent will be obtained by an-1 a that is into changing an-1 into n ba-1 b2-1+bn Hence in (i) we must put a, b, for a for b; we then have n-1 an-1 b bb + a n-1 n n and b b + a n-1 Thus the law will hold good for the (n+1)th convergent if it holds good for the nth convergent. But we know that the law holds good for the third convergent; it must therefore hold good for all subsequent ones. COR. II. In the fraction 8-8-8- 2 Pn=bnPn-1-αn Pn-2 and In=bnIn-1 — AnIn-2' Ex. By means of the law connecting successive convergents to a continued fraction, find the fifth convergent of each of the following fractions: form a + 1 1 1 where a, b, c, d,... are all positive integers, have certain properties on account of which such fractions have special utility: these properties we proceed to consider. We first however shew that any rational fraction can be reduced to a continued fraction of this type with a finite number of elements. m n For let be the given fraction; then, if m be greater than n, divide m by n and let a be the quotient and p the remainder, so that = a + Now divide n by p m n P. n and let b be the quotient and q the remainder; then Now divide p by q and let c be the 1 = n Since the numbers p, q, r,... become necessarily smaller at every stage, it is obvious that one of them will sooner or later become unity, unless there is an exact division at some earlier stage, so that the process must terminate after a finite number of divisions. It should be noticed that the process above described is exactly the same as that for finding the G. C. M. of m and n, the numbers a, b, c,... being the successive quotients. On this account the numbers a, b, c &c. in the 1 1 continued fraction a + b+c+ are often called the first, second, third, &c. partial quotients. It is easy to see that the continued fractions, found as m mk above, for and where k is any integer, will be the same. n Pn_Pn-1 _ an Pn-1+Pn-2 Pn-1 - Pn-29n-1 — Pn-19n-2 ; = In In-1 anIn-1 + In-2 In-1 = InIn-1 PnIn-1-Pn-19=-(Pn-1In-2-Pn-2n-1). |