Re: Series expansion of x_n=Tan[x_n]
- To: mathgroup at smc.vnet.net
- Subject: [mg95892] Re: Series expansion of x_n=Tan[x_n]
- From: Scott Hemphill <hemphill at hemphills.net>
- Date: Thu, 29 Jan 2009 05:55:48 -0500 (EST)
- References: <glpfin$kot$1@smc.vnet.net>
- Reply-to: hemphill at alumni.caltech.edu
Francois at news53rd.b1.woo, Fayard at news53rd.b1.woo writes:
> Hello,
>
> I'm new to Mathematica and I want to comptute a series expansion of the
> sequence (x_n) defined by :
>
> x_n=Tan[x_n] and n Pi-Pi/2 < x_n < n Pi+Pi/2
>
> It's easy to prove that
>
> x_n = n Pi + O(1) and x_n = n Pi + ArcTan[x_n]
>
>>From these 2 formulas, one could easily compte a series expansion of
> (x_n) to any order. For example:
>
> x_n = n Pi + ArcTan[nPi + O(1)] = nPI + Pi/2 -1/(n Pi) + O(1/n^2)
>
> Then we can iterate the Process.
>
> I want to do this whith Mathematica, but I have a Few Problems :
> - How can I enter O(1) ? I've tried O(n,Infinity)^0 but it simplifies to=
> 1
> - When I compute ArcTan[n Pi + Pi/2- 1/(Pi n)+O(1/n)^2), it gives me
> Pi/2-1/(Pi n)+O(1/n)^2. I'm surprised because one could get a better
> serie expansion from that.
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
--
Scott Hemphill hemphill at alumni.caltech.edu
"This isn't flying. This is falling, with style." -- Buzz Lightyear