Convergents Sentence Examples

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.