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: <email@example.com>
- 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
> 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.
f[a] == a+(a+1)/(a+2+(a+3)/(a+4+(a+5)/(a+6+...)))
then you want to compute f. Note that
f[a] == a + (a + 1)/f[a + 2]
RSolve can solve recurrence relations, but it cannot solve this
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, 15]
conv = % /. f -> Identity
then the numerators of the convergents, that is,
1, 5, 29, 233, 2329, 27949, 78257, 6260561, 112690097, ...
and a search for this sequence at
has links to
and from there to
At these last two URLs, the sequence is given in closed form as
Another alternative is to convert the non-simple continued fraction into
a simple continued fraction,
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
which is the continued fraction for square root of E. See also
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)
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