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Analyzing sequences of fractions in Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg93520] Analyzing sequences of fractions in Mathematica
*From*: sergio_r at mail.com
*Date*: Thu, 13 Nov 2008 04:04:30 -0500 (EST)
Hello Mathematica users,
(* I am trying to infer, if possible, the general term
of the following sequence: *)
mysequence = {1/8, 3/16, 11/32, 17/64, 25/128, 59/256, 147/512,
265/1024, 465/2048,
995/4096, 2171/8192, 4161/16384, 7881/32768, 16203/65536,
33571/131072,
65977/262144, 129025/524288, 260979/1048576, 529547/2097152,
1051505/4194304, 2083705/8388608, 4186715/16777216, 8423091/33554432,
16796521/67108864, 33466161/134217728, 67059203/268435456,
134443931/536870912}
(*
This sequence seems to converge to 1/4 as suggested by taking
N[mysequence].
In addition, one can verify that each denominator
is a power of 2 through the command
FactorInteger[ mysequence ]
Thus, one could subtract 1/4 from each term in "mysequence"
obtaining:
mysequenceminus1over4 = mysequence -1/4;
and one could propose that the general term for this new
sequence "mysequenceminus1over4" has the form
a[n_] = f[n]/2^{2*n}
from where one could obtain that
f[n+1] = 2 (a[n+1]/a[n])*f[n]
which leads to
f[m]/f[1] = 2^(m-1)*(a[m]/a[1]) for m=1,2,3,..
Thus, the graphics of the left hand side of this function
could eventually guide us to guess the form of the
form of f[n].
sequenceof2 = Table[2^(m-1),{m,1,Length[mysequenceminus1over4]}];
rhsfunc = sequenceof2*(mysequenceminus1over4/mysequenceminus1over4
[[1]]);
datatoplotfunc = Table[{m,rhsfunc[[m]]},{m,1,Length[rhsfunc]}]
ListPlot[datatoplotfunc]
Unfortunately I don't see a useful form of f[n] from
this plot. Accordingly, I am wondering if there is a general
way to approach
this problem from a different perspective.
Thanks in advance,
Sergio
sergio_r at mail.com
*)
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