# Convergents Sentence Examples

convergents
• The exact determination of the appearances in any given case is a mere problem of convergents to a continued fraction.

• 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.

• 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 infinite general continued fraction of the first class cannot diverge for its value lies between that of its first two convergents.

• 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.

• The following are the main properties of the convergents.