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[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. n (3/4) 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

**References**:**Limit of nested function***From:*Miguel <mibelair@hotmail.com>