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

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    $\begingroup$ 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 $a_n$. $\endgroup$ – Fedor Petrov Aug 1 at 7:34
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    $\begingroup$ @FedorPetrov Indeed, this is, e.g., proposition 2 in these notes (Peter Roquette, “On convergent power series”, 1996-07-16). $\endgroup$ – Gro-Tsen Aug 1 at 10:02
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    $\begingroup$ All series with zero radius of convergence are transcendental over $C(X)$. $\endgroup$ – Alexandre Eremenko Aug 1 at 13:15
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    $\begingroup$ Wahoo. Thanks for these very accurate answers. Roquette's paper is very impressive. $\endgroup$ – joaopa Aug 1 at 18:47

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