CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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$\sum_{n \geq 0} n! x^{n}$ is a formal power series that is not rational over $Q$

Statement refuted

Every formal power series over $Q$ is rational.

The series $F (x) = \sum_{n \geq 0} n! x^{n}$ is a counterexample.

Facts & Assumptions

Given: The formal series $F (x) = \sum_{n \geq 0} n! x^{n} \in Q ⟦ x ⟧$ .

[L1]

A formal series over a field is rational if and only if its coefficient sequence satisfies an eventual constant-coefficient recurrence (A coefficient sequence is eventually linearly recurrent if and only if its formal generating function is rational).

[L2]

A nonzero polynomial of degree $d$ over an integral domain has at most $d$ distinct roots (A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots).

Counterexample

technique · contradiction by polynomial degree

1.1

Suppose $F$ were rational. By [L1] its coefficient sequence would satisfy an eventual constant-coefficient recurrence. That recurrence cannot have order zero, since an eventual order-zero recurrence makes the sequence eventually zero while $n! \neq 0$ in $Q$ for every $n$ . So there would be $d \geq 1$ , coefficients $c_{1}, \dots, c_{d} \in Q$ with $c_{d} \neq 0$ , and an index $N$ such that $(n + d)! + c_{1} (n + d - 1)! + \dots + c_{d} n! = 0$ for every $n \geq N$ .

assume-contraL1

2.1

Divide the relation by the nonzero integer $n!$ . It says that the polynomial $P (z) = (z + 1) \dots (z + d) + c_{1} (z + 1) \dots (z + d - 1) + \dots + c_{d - 1} (z + 1) + c_{d}$ vanishes at every integer $z = n \geq N$ .

step 1.1algebra

3.1

The polynomial $P$ has degree $d$ and leading coefficient $1$ , so it is nonzero. But step 2.1 gives it more than $d$ distinct rational roots, contradicting [L2].

step 2.1L2algebra

4.1

Therefore the coefficient sequence is not eventually recurrent and [L1] shows that $F (x)$ is not rational. This argument is entirely formal and uses no convergence claim.

step 3.1L1discharge-contradiction∎

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∑n≥0n!xn∑n≥0​n!xn is a formal power series that is not rational over QQ