continued fractions

[Richard Guy]

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?)