Optimal error bounds for convergents of a family of continued fractions

Let F be the family of continued fractions K(a(p)/1), where a(1) = -g(1), a(p) = (1 -g(p-1))g(p)x(p), p = 2,3,..., with 0 less than or equal to g(p) less than or equal to 1, g(p) fixed, and \x(p)\ less than or equal to 1, p = 2, 3,.... In this work, we derive upper bounds on the errors in the convergents of K(a(p)/1) that are uniform for F, and optimal in the sense that they are attained by some continued fraction in F. For the special case g(i) = g < 1/2, i = 1, 2,..., this bound turns out to be especially simple, and for g(i) = g = 1/2, i = 1,2,..., the known best form of the theorem of Worpitzki is obtained as an immediate corollary. (C) 1996 Academic Press, Inc.