A formal coordinate need not converge

On $X={\mathbb{P}}^1$ at $p=0$, take the formal series $f(t)=t+{\sum }_{n\ge 1}n!t^{n+1}$. The map ${\tau }_f:\mathbb{C}\mathrm{⟦}z\mathrm{⟧}\to \mathbb{C}\mathrm{⟦}t\mathrm{⟧}$ sending $z\mapsto f(t)$ and fixing constants is a continuous ring isomorphism: its linear coefficient is $1$, so the coefficients of a compositional inverse are determined recursively as in [F2]. By [F3], it is a formal coordinate of the pointed curve.

For any fixed complex $t\mathrm{\ne }0$, the ratio of consecutive term magnitudes in the tail is $(n+1)\mathrm{\mid }t\mathrm{\mid }$, which tends to infinity. In particular, the terms $n!t^{n+1}$ do not tend to zero, so the series diverges at every nonzero $t$. Its radius of convergence is therefore zero, and by [F4] it cannot be the Taylor series of an analytic coordinate germ.