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MathGroup Archive 2006

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Re: Limit of Infinitely Nested Expression (4.0 succeeds, 5.2 fails...)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71680] Re: Limit of Infinitely Nested Expression (4.0 succeeds, 5.2 fails...)
  • From: bobbaillie at frii.com
  • Date: Sun, 26 Nov 2006 03:49:14 -0500 (EST)
  • References: <ek97sj$irp$1@smc.vnet.net>

I contacted Wolfram tech support about this bug.  They replied that 4.0
was in error, and that 5.2 is working as it should. (!)  I don't buy
their argument.  I think they should fix it, to make it possible to
compute the exact limit in this case.  Nevertheless, here is their
reply:

I believe that Mathematica is behaving correctly in
this example.

Nest and NestList allow you to apply functions a fixed number of times.
 In
this sense, the behavior of Ver. 5.X is correct while the behavior of
the
package "Limit" for Ver. 4 is inappropriate.

Often you may want to apply functions until the result no longer
changes.
You can do this using FixedPoint.

  In[2]:=
  FixedPoint[N[Sqrt[5 + #1] &, 40], N[5, 40]]

  Out[2]=
  2.791287847477920003294023596864004244492

Note that, the following won't terminate.

  In[3]:=
  FixedPoint[Sqrt[5 + #1] &, 5]

  Out[3]=
  $Aborted

because "SameTest" is done symbolically.  You may do

  In[4]:=
  FixedPoint[Sqrt[5 + #1] &, 5.0000000000000000]

  Out[4]=
  2.791287

though it will take a lot more time if you add more "0"'s to the right
of

  5.0000000000000000

This is again because of symbolic "SameTest."

If you do want to symbolically compute the fixed point of the function,
then, as you pointed out, you have to manually solve the equation:

  In[5]:=
  Solve[x == Sqrt[5 + x]]

  Out[5]=
	 1 + Sqrt[21]
  {{x -> ------------}}
	      2


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