The transcendence of $\mathrm{e}$ via formal power series

We review Hilbert's classical analytical proof of the transcendence of the number $\mathrm{e}$. Then, we show how this result can be obtained algebraically by means of formal power series (FPS). We give two proofs of the transcendence of $\mathrm{e}$ based on FPS. The first of them is a specialization of the 1990 proof by Beukers, B\'ezivin and Robba of the Lindemann-Weierstrass theorem. The second proof is due to this author and is an adaptation of Hilbert's argument to FPS.