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Re: Limit of non-simple continued fraction
*To*: mathgroup at smc.vnet.net
*Subject*: [mg64710] Re: Limit of non-simple continued fraction
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Tue, 28 Feb 2006 05:02:14 -0500 (EST)
*Organization*: The University of Western Australia
*References*: <dtrvmt$lv0$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <dtrvmt$lv0$1 at smc.vnet.net>,
Sinan Kapçak <sinankapcak at yahoo.com> wrote:
> i want to find the value of the limit
>
> 1+(2/(3+(4/(5+(6/(7+...
>
> how can i do that with Mathematica?
As far as I'm aware, there is no built-in tool for computation of
non-simple continued fractions in Mathematica.
Defining
f[a] == a+(a+1)/(a+2+(a+3)/(a+4+(a+5)/(a+6+...)))
then you want to compute f[1]. Note that
f[a] == a + (a + 1)/f[a + 2]
RSolve can solve recurrence relations, but it cannot solve this
nonlinear one.
If you compute the convergents to 1+2/(3+4/(5+6/7+...), using the above
recurrence as a replacement rule,
NestList[# /. f[a_] :> a + (a + 1)/f[a + 2] &, f[1], 15]
conv = % /. f -> Identity
then the numerators of the convergents, that is,
Numerator[conv]
are
1, 5, 29, 233, 2329, 27949, 78257, 6260561, 112690097, ...
and a search for this sequence at
http://www.research.att.com/~njas/sequences
has links to
http://www.research.att.com/~njas/sequences/A113012
http://www.research.att.com/~njas/sequences/A113011
and from there to
http://mathworld.wolfram.com/ContinuedFraction.html
At these last two URLs, the sequence is given in closed form as
1/(Sqrt[E]-1).
Another alternative is to convert the non-simple continued fraction into
a simple continued fraction,
ContinuedFraction[conv]
The pattern is
1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, 1, 21, 1, 1, 25, 1, 1, ...
A search on this pattern finds it as
http://www.research.att.com/~njas/sequences/A058281
which is the continued fraction for square root of E. See also
http://www.numbertheory.org/php/davison.html
and
http://www.research.att.com/~njas/sequences/Sindx_Con.html#confC
which is the index of entries for continued fractions for constants.
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
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