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