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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)    
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