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I would like to find a two-term or a three-term asymptotic expansion of $x_{n}$ the unique solution of $$x_{n}=\frac{1}{\tan(x_{n})}$$ on the interval $]n\pi,n\pi+\pi[ $

We have: $$ x_{n}=n\pi+\arctan(\frac{1}{x_{n}})$$

So $$x_{n} \sim_{n\rightarrow \infty} n\pi$$

What is the method to find the next terms of the asymptotic expansion?

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    $\begingroup$ $\arctan(z)=z+o(z)$ when $z\to0$ hence $x_n=n\pi+1/(n\pi)+o(1/n)$. $\endgroup$ – Did May 28 '12 at 11:51
  • $\begingroup$ Thanks! I should have thought more before asking... $\endgroup$ – Chon May 28 '12 at 14:12

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