The law of relation of successive **convergents** to a continued fraction involves more advanced methods (see ï¿½ 42 (iii.) and Continued Fraction).

(iii.) Another application of the method is to proving the law of formation of consecutive **convergents** to a continued fraction (see Continued Fractions).

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.