LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The exponential series converges absolutely for every real argument

Statement

For every real $x$ , the series $\sum_{n \geq 0} x^{n} / ι (n!)$ converges absolutely. Its power-series radius is therefore $+ \infty$ .

Facts & Assumptions

Given: A real $x$ .

[L1]

Archimedes supplies a natural larger than any prescribed real (Every complete ordered field is Archimedean).

[L2]

A tail bounded termwise by a convergent geometric series converges (If $0 \leq a_{k} \leq b_{k}$ eventually, convergence of $\sum b_{k}$ gives convergence of $\sum a_{k}$ , and divergence of $\sum a_{k}$ gives divergence of $\sum b_{k}$ , For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges), and absolute convergence implies convergence (If $\sum ∣ a_{k} ∣$ converges then $\sum a_{k}$ converges).

[L3]

Factorials satisfy $(n + 1)! = n! (n + 1)$ and are nonzero naturals; every positive natural has a positive, hence nonzero, canonical real image (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, Canonical naturals are positive and strictly increasing).

Proof

technique · direct

1.1L1L3choose

If $x = 0$ , the series is $1 + 0 + 0 + \dots$ and converges absolutely. Hence assume $x \neq 0$ . Choose $N \geq 1$ with $ι (N) > 2 ∣ x ∣$ . For $n \geq N$ , the absolute terms $a_{n} = ∣ x ∣^{n} / ι (n!)$ are positive and satisfy $a_{n + 1} / a_{n} = ∣ x ∣ / ι (n + 1) < 1 / 2$ .

2.1step 1.1L2

Thus $a_{N + j} \leq a_{N} 2^{- j}$ by induction, and the tail is dominated by a convergent geometric series.

3.1step 1.1step 2.1L2∎

The zero case from step 1.1 and, when $x \neq 0$ , adding the finite initial segment to the convergent tail prove absolute convergence for arbitrary $x$ . Hence every nonnegative radius works and the radius is $+ \infty$ .

The exponential series converges absolutely for every real argument

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources