Formal multiplicative group (${\hat{\mathbb{G}}}_m$)

The formal multiplicative group has law $F_m(X,Y)=X+Y+XY$, obtained by writing a multiplicative coordinate as $1+X$. Its multiplication-by-$n$ series is

Over an algebra over a field whose scalar field is $\mathbb{Q}$, the series $\exp (X)-1$ defines an isomorphism from the formal additive group to the formal multiplicative group, with inverse $\log (1+X)$.