# Transcendence of Euler series

Is the Euler series $$\sum_{n\ge0}n!X^n\in\mathbb C[[X]]$$ transcendental over $$\mathbb C(X)$$?

• Does not any algebraic power series has a positive radius of convergence? It looks that the bound $C^n/n^D$ for the coefficients should be provable by induction, looking at the relation for the new coefficient $a_n$. – Fedor Petrov Aug 1 at 7:34
• @FedorPetrov Indeed, this is, e.g., proposition 2 in these notes (Peter Roquette, “On convergent power series”, 1996-07-16). – Gro-Tsen Aug 1 at 10:02
• All series with zero radius of convergence are transcendental over $C(X)$. – Alexandre Eremenko Aug 1 at 13:15
• Wahoo. Thanks for these very accurate answers. Roquette's paper is very impressive. – joaopa Aug 1 at 18:47