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
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