Optimal error bounds for convergents of a family of continued fractions

Let ℱ be the family of continued fractions K(a_p/1), where a₁ = -g₁, a_p = (1 - g_p-1) g_pX_p, p = 2,3,..., with 0 ≤ g_p ≤ 1, g_p fixed, and |x_p| ≤ 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.