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Re: Limit of nested function

On 10/16/2011 03:45 PM, Miguel wrote:
> How can I to calculate the limit of a nested function . Mathematica 6
> yields an error message "... Non negative machine size integer ...".
> For example, let
> f[x,n]=Nest[Sqrt[# Sqrt[#]]&,x,n]
> For x=3,      Limit[f[3,n], n->inf]
> Thanks

One possibility is to solve the recurrence with an appropriate initial 
condition, and take the limit of that as n->Infinity.

In[26]:= gn = g[n] /. RSolve[{g[n+1]==Sqrt[g[n]*Sqrt[g[n]]], g[0]==3},
   g[n], n]
Solve::ifun: Inverse functions are being used by Solve, so some 
solutions may
      not be found; use Reduce for complete solution information.
Out[26]= {3      }

In[27]:= Limit[gn, n->Infinity]
Out[27]= {1}

So that limit is 1.

Another way is to solve for teh fixed points, then figure out 
numerically which is the one you want.

f[x_,n_] := Nest[Sqrt[# Sqrt[#]]&,x,n]

In[29]:= ffxtpt = x/. Solve[f[x,1] == x, x]
Out[29]= {0, 1}

In[30]:= N[f[3,10]]
Out[30]= 1.06382

This again indicates the limit is 1.

Daniel Lichtblau
Wolfram Research

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