       Re: Limit of nested function

• To: mathgroup at smc.vnet.net
• Subject: [mg122176] Re: Limit of nested function
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Tue, 18 Oct 2011 07:42:08 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201110162045.QAA19694@smc.vnet.net>

```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:= gn = g[n] /. RSolve[{g[n+1]==Sqrt[g[n]*Sqrt[g[n]]], g==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.
n
(3/4)
Out= {3      }

In:= Limit[gn, n->Infinity]
Out= {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:= ffxtpt = x/. Solve[f[x,1] == x, x]
Out= {0, 1}

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

This again indicates the limit is 1.

Daniel Lichtblau
Wolfram Research

```

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