CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Substituting $1$ into $1 + x + x^{2} + \dots$ is not a defined formal composition

Counterexample

Over $Z$ , let $f = 1 + x + x^{2} + \dots$ . The formal expression $f \circ 1$ is not defined.

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

Formal composition is $f \circ g = \sum_{n \geq 0} [x^{n}] f g^{n}$ , defined when $f$ is a polynomial or when $g (0) = 0$ (Composition $f \circ g$ of formal series when the outer series is a polynomial or the inner series has zero constant term).

Verification

technique · test local finiteness in degree zero

1.1

The outer series $f$ is not a polynomial, and the inner series $1$ does not have zero constant coefficient, so neither admissibility branch applies. More concretely, the proposed constant coefficient would be $\sum_{n \geq 0} 1$ , an infinite sum not defined by the ring operations of $Z$ .

givenF1

1.2

By contrast, $x^{2}$ has zero constant coefficient, so $f \circ x^{2} = 1 + x^{2} + x^{4} + \dots$ is defined; every coefficient has at most one contributor.

givenF1

2.1

Hence $f \circ 1$ is undefined as a formal composition, while the zero-constant substitution in step 1.2 is admissible. This is a local-finiteness obstruction, not a claim about analytic divergence.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Benjamin Sambale, An Invitation to Formal Power Series (standard reference, not scraped)

Substituting 11 into 1+x+x2+⋯1+x+x2+⋯ is not a defined formal composition