Another example of a sequence is afforded by the successive convergents to a continued fraction of the form ao+ I I, al+ a2+ï¿½ where ao,a 1, a 2, ...
Denoting these convergents by Po/Qo, P1/Q1, P2/Q2, ...
The exact determination of the appearances in any given case is a mere problem of convergents to a continued fraction.
A2 a3 a 4 Ia2 la3 la4 The terminating continued fractions b 2 b2 b 3 b, b, b4 al, a1 + ?2, al-1 7:2+Z' al' + a2 reduced to the forms ala2a3+b2a3+ b2a1 a2 a2a3 b3 ala2a3a4+b2a3a4+b3a,a4+b4a,a2 +b2b4 a2a,a4+a4b3+a2b4 ' are called the successive convergents to the general continued fraction.
The numerators and denominators of the successive convergents obey the law p n g n _ l - pn-1qn = (- O n, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property.
Every simple continued fraction must converge to a definite limit; for its value lies between that of the first and second convergents and, since f ?n _ _1 I, L t.
Of which the successive convergents are 3 22 333 355 103993 &c., I' 7' 106' 113' 33102' the fourth of which is accurate to the sixth decimal place, since the error lies between I /g4g5 or.
If we suppose alb 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.
- The numerators and denominators of the convergents to the general continued fraction both satisfy the difference equation un =anu„_,+bnun_2.
In fact, a continued fraction ai +a2+ +an+ can be constructed having for the numerators of its successive convergents any assigned quantities pi, P2, P3,
John Wallis, discussing this fraction in his Arithmetica finitorum (1656), gives many of the elementary properties of the convergents to the general continued fraction, including the rule for their formation.
, obtained by taking account of the successive quotients, are called convergents, i.e.
Convergents to the true value.
(i) If we precede the series of convergents by i and - 1 6 -, then the numerator (or denominator) of each term of the series o i a, ab?-1 after the first two, is found by multiplying 1, o?
(iii) The convergents are alternately less and greater than the true value.
(v) The difference of two successive convergents is the reciprocal of the product of their denominators; e.g.
Bc-+1 b b(bc-}-I) It follows from these last three properties that if the successive convergents are pi P2 3,..