<s>[INST] Power (Laurent) Series of $\coth(x)$ I need some help to prove that the power series of $\coth x$ is:
$$\frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} + O(x^5) \ \ \ \ \ $$
I don't know how to derive this, should I divide the expansion of $\cosh(x)$ by the expansion of $\sinh(x)$? (I've tried but without good results :( )
Or I have to use residue calculus? 
Anyone can suggest me a link where I can find a detailed explanation of this expansion?
Thanks.
 [/INST] Long division of series with $\cosh(x) = 1 + \dfrac{x^2}{2} + \dfrac{x^4}{24} + \ldots$
and $\sinh(x) = x + \dfrac{x^3}{6} + \dfrac{x^5}{120} + \ldots$.  Unfortunately I don't know how to typeset this nicely in LaTeX.
First term is $1/x$, 
$$1 + \dfrac{x^2}{2} + \dfrac{x^4}{24} + \ldots - \dfrac{1}{x} \left(x + \dfrac{x^3}{6} + \dfrac{x^5}{120} + \ldots\right) = \frac{x^2}{3} + \frac{x^4}{30} + \ldots$$
Next term is $x/3$,
...
</s>