LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The exponential series converges absolutely for every real argument

Statement

For every real $x$ , the series $\sum_{n\ge0}x^n/\iota(n!)$ 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 $0 \le a_k \le b_k$ 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 $|r| < 1$ , $\sum_{k \ge 0} r^k = 1/(1-r)$ , and for $|r| \ge 1$ 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 $\mathbb{N}$ , The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Canonical naturals are positive and strictly increasing).

Proof

technique · direct

1.1

If $x=0$ , the series is $1+0+0+\cdots$ and converges absolutely. Hence assume $x\ne0$ . Choose $N\ge1$ with $\iota(N)>2|x|$ . For $n\ge N$ , the absolute terms $a_n=|x|^n/\iota(n!)$ are positive and satisfy $a_{n+1}/a_n=|x|/\iota(n+1)<1/2$ .

L1L3choose

2.1

Thus $a_{N+j}\le a_N2^{-j}$ by induction, and the tail is dominated by a convergent geometric series.

step 1.1L2

3.1

The zero case from step 1.1 and, when $x\ne0$ , adding the finite initial segment to the convergent tail prove absolute convergence for arbitrary $x$ . Hence every nonnegative radius works and the radius is $+\infty$ .

step 1.1step 2.1L2∎

Depends on

Used by

Cited to discharge well-definedness by The real exponential function and the number e by a power series.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 102 results over 22 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

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)