Expansion of the hyperbolic cotangent

Question

I have this question:
Prove that $$\coth(z) = \sum_{n=0}^\infty {B_{2n} \over 2(2n)!}(2z)^{2n−1}$$ $\forall |z| < π$.
I have already obtained the expression, but I don't know how to get the |z| < π. I have looked it up, and you can use the Asymptotic Formula for Bernoulli numbers, but this is an exercise for class and the teacher hasn't given us this formula, so we can't use it. I would like to know if there's any other method to know why the expression only works if |z| < π

Answer 1

By definition, $\operatorname{coth}(z)=\frac{\cosh(z)}{\sinh(z)}$. For each $z\in\Bbb C\setminus\{\pm in\pi\mid n\in\Bbb N\}$, let$$f(z)=\begin{cases}z\operatorname{cosh}(z)&\text{ if }z\ne0\\1&\text{ if }z=0,\end{cases}$$which is an analytic function. Its Taylor series centered at $0$ converges on any disk centered at $0$ contained on its domain. In particular, it converges on $D(0,\pi)$ and therefore the Laurent series of $\operatorname{coth}(z)\left(=\frac{f(z)}z\right)$ also converges on that disk.