Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)

MÁTYÁS, F., Linea r recurrences and rootfinding methods

Linear recurrences and rootfinding methods 33 Proof of Theorem 2. To prove the theorem we have to show that R^

d - a

d lim —

jfr = 0. n—foo n(

k) nd n,d a Applying (4) and (3), we get that n,d a _ )

Vn,d a n ,0 v

vn, 0 fí(

k) r

vd ~ W^ - rvdW^ W^ n,d Wn,d Q n ,0 l

tn,0 ,-, „(,-,)!(I)'(f) l-(L)>(t

n' from which, by |cvj > \ß\ and / > k > 1, >( 0 R lim —

ttt — — 0 n,d follows. References [1] JAMIESON, M. J., Fibonacci numbers and Aitken sequences revisited, Amer. Math. Monthly, 97 (1990), 829-831. [2] MÁTYÁS, F., Recursive formulae for special continued fraction convergents, Acta Acad. Paed. Agriensis Sect. Mat., 26 (1999), 49-56. [3] MCCABE J. H. AND PHILLIPS, G. M., Aitken sequences and generalized Fibonacci numbers, Math. Comp., 45 (1985), 553-558. [4] MUSKAT, J. B., Generalized Fibonacci and Lucas sequences and rootfinding methods, Math. Comp., 61 (1993), 365-372. [5] PHILLIPS, G. M., Aitken sequences and Fibonacci numbers, Amer Math. Monthly, 91 (1984), 354-357. [6] TAHER R. B. AND RACHIDI, M., Application of the e- algirithm to the ratio of »'-generalized Fibonacci sequences, The Fibonacci Quarterly, 39 (2001), 22-26. [7] ZHANG, Z., A class of sequences and the Aitken transformation, The Fibonacci Quarterly , 36 (1998), 68-71.