New convergence criterion for vector valued continued fractions
Xiao Ping · Journal of Hefei University of Technology · 2002
A new approach to prove the convergence properties of continued fractions is given,which shows that the classical backward algorithm is an efficient tool in the convergence theory of continued fractions. In the paper,the backward algorithm in the scalar case is generalized to the vector case, and a recurrence relation for the difference between two consecutive convergents of vector continued fractions is obtained. Then, by means of the relation, an analogy of Pringsheim convergence criterion is proved for vector continued fractions of the form K (a n/b n),where b n satisfies Smelson inverse. Finally, a truncation error estimation is given.