CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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An infinite family of constant series $1$ is not summable in the formal topology

Counterexample

The family $(f_{i})_{i \in N}$ with $f_{i} = 1$ for every $i$ is not summable in $R ⟦ x ⟧$ for any nonzero commutative ring $R$ .

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

A family is summable exactly when, below every degree cutoff $N$ , only finitely many members have a nonzero coefficient (Summable families of formal series are locally finite in every coefficient range).

[F2]

For $f = \sum a_{n} x^{n}$ , coefficient extraction is evaluation: $[x^{n}] f = a_{n}$ (Formal power series over a commutative ring and the coefficient-extraction functional $[x^{n}]$ ).

Verification

technique · inspect the first coefficient range

1.1

Take $N = 1$ . Every index $i$ contributes the nonzero coefficient $[x^{0}] f_{i} = 1$ , so infinitely many family members have a nonzero coefficient below $N$ .

givenF1F2

1.2

In contrast, the family $(x^{n})_{n \geq 0}$ is summable: below any fixed degree $N$ , only the indices $n < N$ contribute. Its coefficientwise sum is the series with every coefficient $1$ .

givenF1

2.1

Step 1.1 violates the defining local-finiteness condition, whereas step 1.2 satisfies it. The nonzero-ring hypothesis is necessary: in the zero ring, the constant series $1 = 0$ and the original family is summable.

step 1.1step 1.2givenF1F2∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 9 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.

An infinite family of constant series 11 is not summable in the formal topology