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Re: Limit of nested function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122171] Re: Limit of nested function
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Tue, 18 Oct 2011 07:41:14 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <j7ffsb$j8b$1@smc.vnet.net>

Hi.
What is beautiful with this forum is that apart of improving his
Mathematica skills, one can
improve deeply his mathematical understanding!
There are persons here like Daniel Lichtblau and Andrzej Kozlowski (to
name just a few!) that their
answers are like complete lectures!
Also most times when you ask something, somebody has asked something
similar in the past!!

See this link from an old post that arose from a similar question of
mine, back in 2006.

http://groups.google.com/group/comp.soft-sys.math.mathematica/browse_thread/thread/b33bb33fc86f296c/9fa4d93bc30ef9f7?lnk=gst&q=limit+of+nested+function#9fa4d93bc30ef9f7

You will find it quite useful.

Let's see therefore your nested function.

"The limit of a nested expression of the form: Nest[f,...], if it
exists, is a fixed point of the function f. A fixed point of f is a
root of f[x]-x==0. "

In[12]:=
Solve[(x == Sqrt[#1*Sqrt[#1]] & )[x]]
Out[12]=
{{x -> 0}, {x -> 1}}

In[30]:=
Take[Table[N[Nest[Sqrt[#1*Sqrt[#1]] & , 3, n], 100], {n, 2, 500}], -4]
Out[30]=
{1.0000000000000000000000000000000000000000000000000000000000000088367457204909355234061713384536523777,
1.0000000000000000000000000000000000000000000000000000000000000066275592903682016425546285038402392833,
1.0000000000000000000000000000000000000000000000000000000000000049706694677761512319159713778801794625,
1.0000000000000000000000000000000000000000000000000000000000000037280021008321134239369785334101345968}

In[35]:=
FixedPoint[N[Sqrt[#1*Sqrt[#1]] & , 100], N[3, 100], SameTest ->
(Abs[#1 - #2] < 1/10000000000 & )]
Out[35]=
1.0000000002633672286467555600009138543119629410021971435246219138655285754491\
500282111126864472584129519268862

In Daniel Lichtblau's words (from above mentioned link):
"Limit is designed to work on functions defined on a continuum, and
known
in a closed form. The function in question satisfies neither
requirement...
Bottom line is that the Limit function was not designed or intended to
handle functions defined on only a sequence of points, or to handle
functions not given in explicit closed form...For examples such as
this one gets results exactly... using Solve to find an exact
representation of the fixed point for the iteration."

Dimitris Anagnostou



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