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Series expansion of x_n=Tan[x_n]


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=
 - 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 ?


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