Continued fraction problem

• To: mathgroup at smc.vnet.net
• Subject: [mg17039] Continued fraction problem
• From: "Alan W.Hopper" <awhopper at hermes.net.au>
• Date: Wed, 14 Apr 1999 02:11:53 -0400
• Sender: owner-wri-mathgroup at wolfram.com

``` A quite artificial but nevertheless interesting construction
in recreational maths is the Champernowne Constant, defined
in Eric Weisstein's Concise Encyc. of Math.,
as the decimal 0.1234567891011... , obtained by concatenating
the positive integers.

The continued fraction expansion of this  number is stated to
be ;  (0,8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15, a 166 digit #,...,
(position 41)a 2504 digit #,...,(position 163)a 33102 digit #,...

And using this code for the continued fraction period in list form ,

In[1]:= cf[x_Real, n_]:=Module[{ip, fp=x, result = {}},
Do[ip = Floor[fp];
AppendTo[result, ip];
fp = 1/(fp-ip), {n}];
result];

In[2]:= cf[0.12345678910111213141516171819202122,20]

Out[3]=
(Error messages detailing Indeterminate,
Infinite expressions and ComplexInfinity.)

{0,8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15,1787142709274,
ComplexInfinity}

All correct up to where the 166 digit term should appear.
(Inserting ,  N[....., 200] in the code does not help).

The result above is better however than with ;

<<NumberTheory`ContinuedFractions`

ContinuedFraction[0.123456789101112131415161718192021, 20]

which gives 150571 as the fifth term.

I suppose that rounding errors and inadequate precision are coming
into play with this situation as it is now.

So I wonder if anyone knows a workable method of obtaining the correct
continued fraction period of a difficult case like the Champernowne const.
with Mathematica, say up to that enormous 163 rd term, if it is at all
feasible?

Alan Hopper
awhopper at hermes.net.au

```

• Prev by Date: Re: shading in area
• Next by Date: RE: SetDirectory Problems