CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-07-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The series $\sum_{n \geq 0} ι (n!) x^{n}$ converges only at $x = 0$ and has radius zero

Statement

The real power series $\sum_{n \geq 0} ι (n!) x^{n}$ has radius $0$ : it converges at its centre $x = 0$ and diverges at every $x \neq 0$ .

Facts & Assumptions

Given: Coefficients $a_{n} : = ι (n!)$ .

[L1]

Factorials eventually exceed every fixed geometric progression (For every real $x$ , $x^{k} / k! \to 0$ , The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ , The canonical natural $ι (n) = n \cdot 1_{F}$ of a field).

[L2]

Cauchy–Hadamard assigns radius $0$ when the coefficient-root limit superior is $+ \infty$ (Cauchy–Hadamard: the reciprocal radius is ${lim sup}_{k \to \infty} ∣ a_{k + 1} ∣^{1 / (k + 1)}$ , with the zero and infinite cases included).

Verification

technique · direct

1.1

For every real $M > 0$ , [L1] applied to a geometric base larger than $M$ shows that $ι ((k + 1)!)^{1 / (k + 1)} > M$ eventually. Hence the coefficient-root limit superior is $+ \infty$ .

2.1

By [L2], the radius is $0$ . At $x = 0$ only the constant term survives, while for $x \neq 0$ the terms fail to tend to zero, so the stated convergence set follows.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

44 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Cauchy-Hadamard theorem, Encyclopedia of Mathematics (standard reference, not scraped)
Power series, Encyclopedia of Mathematics (standard reference, not scraped)

The series ∑n≥0ι(n!)xn∑n≥0​ι(n!)xn converges only at x=0x=0 and has radius zero