## Convergents 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. - 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,..