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