Seeking help in conducting calculations

[Alexander Povolotsky]

Dear Max,

> It is well known that partial quotients of a continued fraction x tend
> to the value x from both sides.

In those two cases, which I was looking at,
there were no in advance known "x", because instead the consecutive
Engel's expansion coefficients of Engel's expansion of "y" were used
to form the continued fraction.

My posting question point was asking to help to see what the value "x" will be
(possibly in relation to "y").

The later reply from Dr Ron Knott shows the computational resource,
which Ron has and which I could use, to get the answer to my question
- Thank you, Dear Ron.

Regards,
AP

On 2/19/08, Max Alekseyev <maxale at gmail.com> wrote:
> On Feb 19, 2008 4:24 PM, Alexander Povolotsky <apovolot at gmail.com> wrote:
>
> > Below are my  partial results when I tried to use OEIS's A006784 "Pi -
> > 3" Engel's expansion coefficients:
> >
> > 8, 8, 17, 19, 300, 1991, 2492, 7236, 10586, 34588, ...
> >
> >  to form a continued fraction expansion and to see to what value it comes to.
> >
> > I noticed that after few steps the results are showing two
> > sequentially interchanging (and possibly slowly converging to each
> > other) trends:
> >
> > 0.529411765->0.517267552->0.508117341->0.5021273->...
> >
> > and
> >
> > 0.152941176->0.158526794->0.16041072->0.160983247->...
>
> Nothing is surprising here. If you take just *any* sequence of
> positive integers and form a continued fraction, you will see exactly
> the same behavior.
>
> It is well known that partial quotients of a continued fraction x tend
> to the value x from both sides.
> See formula (35) and (36) at
> http://mathworld.wolfram.com/ContinuedFraction.html
>
> Regards,
> Max
>