Seeking help in conducting calculations

[Max Alekseyev]

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