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