#### Sentence Examples

• 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.