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Re: Series expansion of x_n=Tan[x_n]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg95936] Re: Series expansion of x_n=Tan[x_n]
  • From: Francois Fayard <fayard.prof at gmail.com>
  • Date: Fri, 30 Jan 2009 05:44:11 -0500 (EST)
  • References: <glpfin$kot$1@smc.vnet.net> <gls1va$hkl$1@smc.vnet.net>

Hello,

Thanks for the help. It was very helpful. I've discovered the
FixedPoint function that seems to be usefull.

Thanks
Francois

> I wouldn't try entering O[1] or O[1/n] because I haven't found a
> useful interaction between that and Series[].
>
> Your basic iteration can be defined this way:
>
> In[1]:= iter[n_,x_] := n*Pi + ArcTan[x]
>
> In[2]:= iter[n,Infinity]
>
>         Pi
> Out[2]= -- + n Pi
>         2
>
> This is the series you are looking for, with terms up through the
> constant term.
>
>
> In[3]:= iter[n,%]
>
>                       Pi
> Out[3]= n Pi + ArcTan[-- + n Pi]
>                       2
>
> In[4]:= Series[%,{n,Infinity,1}]
>
>                Pi    1       1 2
> Out[4]= Pi n + -- - ---- + O[-]
>                2    Pi n     n
>
> Now this series includes the (1/n) term.
>
>
> In[5]:= Normal[%]
>
>            1      Pi
> Out[5]= -(----) + -- + n Pi
>           n Pi    2
>
> In[6]:= iter[n,%]
>
>                        1     Pi
> Out[6]= n Pi - ArcTan[---- - -- - n Pi]
>                       n Pi   2
>
> In[7]:= Series[%,{n,Infinity,2}]
>
>                Pi    1        1        1 3
> Out[7]= Pi n + -- - ---- + ------- + O[-]
>                2    Pi n         2     n
>                            2 Pi n
>
> Now this series includes the (1/n^2) term.
>
> This whole operation can be put together into one expression:
>
> In[8]:= f[k_] := FixedPoint[Simplify[Series[n*Pi + ArcTan[Normal[#1=
]],
>         {n, Infinity, k}]] & , Infinity]
>
> In[9]:= f[4]
>
>                                              2         2
>                Pi    1        1      8 + 3 Pi    8 + Pi       1 5
> Out[9]= Pi n + -- - ---- + ------- - --------- + -------- + O[-]
>                2    Pi n         2        3  3       3  4     n
>                            2 Pi n    12 Pi  n    8 Pi  n
>
> Scott




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