Radius of convergence of the p-adic exponential
= Radius of convergence of the p-adic exponential
{title2=$p^{-1/(p-1)}$}
On $\mathbb C_p$, the power series $\exp(x)=\sum_{n\geq0}x^n/n!$ has radius of convergence
$$
p^{-1/(p-1)}.
$$
Indeed, <Legendre formula> gives $v_p(n!)/n\to1/(p-1)$, and the <Cauchy-Hadamard theorem> gives the stated radius.