Worpitzky's Theorem on continued fractions

It is known that under the hypotheses in Worpitzky's Theorem on continued fractions, the approximants converge to the value of the continued fraction with error O(1/n), and that this estimate is best possible. Using a geometric argument we give a more refined estimate of the rate of convergence. We also extend a result of Waadeland that is closely connected to Worpitzky's Theorem.