TR#:  CS0769 
Class:  CS 
Title:  OPTIMAL EXTENSIONS OF THE
THEOREM OF WORPITZKY FOR CONTINUED FRACTIONS. 
Authors:  Y. Shapira, A. Sidi and M. Israeli 
CS0769.pdf  
Abstract: 
One of the best known results in the convergence analysis of continued fractions K(a_n/1) is the theorem of Worpitzky. This theorem states that for a_n \leq 1/4, n = 1,2,..., convergence takes place and the approximants w_n, n = 1,2,..., and the limits w of the continued fractions are in the circular region z \leq 1/2. In this work we prove new results under the condition a_n \leq r/4,\ n = 1,2,...,\ 0 < r \leq 1, that are also optimal. In particular, we show that the approximants and the limits are in the circular region z \leq (1\sqrt{1r})/2, and that the errors w_n  w satisfy w_nw \leq v_nv, where v_n, n = 1,2,..., and v are the approximants and limit, respectively, of the continued fraction K(a_n/1) for which a_n =  r/4, n = 1,2,...\ . The theorem of Worpitzky follows from this result by letting r = 1. Another consequence is that, for 0 < r < 1, W_N  W = O([(1\Sqrt{1R})/(1+\Sqrt{1R})]^N) at least, as n \rightarrow \infty.

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