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The problem is that it is not true that the canonical map $i:\mathbb{Z}[[x]]\otimes\mathbb{Q}\to\mathbb{Q}[[x]]$ is an isomorphism. Any power series of the image of $i$ must have only finitely many different denominators on its coefficients, since an element of $\mathbb{Z}[[x]]\otimes\mathbb{Q}$ is a sum involving only finitely many elements of ...

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