Continued fractions of transcendental numbers

It is well known that if, in a continued fraction there is a subsequence of the a n which increases very rapidly, then ξ is a transcendental number. A result of this kind follows from Liouville's Theorem on rational approximations to algebraic numbers, but the most precise result so far established is that which was deduced from Roth's Theorem by Davenport and Roth [1]. They proved (Theorem 3) that if ξ is algebraic, then where q n is the denominator of the n th convergent to (1). Thus if ξ is transcendental.