Open letter to AB, copied to sequensters & munsters, in case they can throw more light (and, in the former case, accept some new sequences). When I asked Hugh Williams, he immediately had the same reaction as I, that it didn't much matter if they were rats or reals, and that a sufficient condition was that all but a finite number of the partial quotients should be greater than one. However, I think it suffices that they be bounded away from zero. Have you looked in a large tome by H.S.Wall -- Continued Fractions ? I haven't. I'll assume they're all positive. If they're periodic, then it seems that the cf converges to the root of a quadratic, whose coefficients are in the field generated by the partial quotients. As a simple example, let's take c_0 = 0, c_n = 1/2 (n>0) See a paper by Bremner & Tzanakis. The convergents are 2/1, 2/5, 10/9, 18/29, 58/65, 130/181, where the numerators and denominators satisfy the recurrence a_n = a_n-1 + 4a_n-2 the denominators are A006131 in OEIS and the numerators twice that (tho this double, 2, 2, 10, 18, 58, 130, ..., seems not to be in OEIS). The cf converges to something with sqrt(17) in it. While not of immediate interest here, I note that c_n = n gives a sequence of convergents whose numerators are duplicated in OEIS as A001053=A103736 (and whose denominators are A001040). If the partial quotients tend to zero, then much of the classical theory still holds, though numbertheoretic aspects tend to disappear. The sequence of convergents either converges or oscillates boundedly. In the following examples I leave the reader to decide which. c_0 = 0, c_n = 1/n 1/1, 1/3, 7/9, 19/45, 159/225, 729/1575, 7407/11025, 48231/99225, ... I've cheated here and not put these in their lowest terms, so that I can say that the numerators and denominators satisfy the recurrence a_n = a_n-1 + n(n-1)a_n-2 The denominators are A000246 in OEIS, but this connexion is not mentioned. The numerators are NOT in OEIS (modulo my not being a good looker). Nor are the numerators or denominators of the convergents to the cf with c_0 = 0 and c_n = 1/2^n: 0/1, 2/1. 2/9, 66/41, 322/1193, 34114/22185, 693570/2465449, ... which satisfy a_n = a_n-1 + 2^2n-1 a_n-2 or the closely related c_n = 1/2^n-1 1/1, 1/3, 9/11, 41/107, 1193/1515, 22185/56299, 2465449/3159019, ... Best to all, R. On Thu, 17 Feb 2005, Andrew Bremner wrote: ...., where can

I find out theorems about the convergence of

cf = c_0 + 1/c_1 + 1/c_2 +1/c_3+...

where c_i are rationals? Specifically, when does cf converge (to a real number?)