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Series expansion of x_n=Tan[x_n]
*To*: mathgroup at smc.vnet.net
*Subject*: [mg95822] Series expansion of x_n=Tan[x_n]
*From*: Francois at news53rd.b1.woo, Fayard at news53rd.b1.woo
*Date*: Wed, 28 Jan 2009 06:29:48 -0500 (EST)
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.
Could you help me ?
Thanks
Francois
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