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

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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.
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

1.1

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 $a_{n + 1} / a_{n} = ∣ x ∣ / ι (n + 1) < 1 / 2$ .

L1L3choose

2.1

Thus $a_{N + j} \leq a_{N} 2^{- j}$ 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 $x \neq 0$ , adding the finite initial segment to the convergent tail prove absolute convergence for arbitrary $x$ . Hence every nonnegative radius works and the radius is $+ \infty$ .

step 1.1step 2.1L2∎

Depends on

Used by

Hadamard instability despite analytic solvability Counterexample
Matrix exponential as the Lie-group exponential Example
The exponential map is surjective on every connected Lie group False statement
The exponential map of a Lie group is a group homomorphism False statement
Local convergence of the Baker–Campbell–Hausdorff series Lemma
Right-trivialized differential of the Lie-group exponential Lemma
The complex exponential series converges absolutely for every complex argument Lemma
Baker–Campbell–Hausdorff theorem Theorem
For every real x, (1+x/n)ⁿ→exp x Theorem
Morse stability with explicit parameter dependence Theorem
Picard iteration from 1 produces the exponential partial sums Theorem
The exponential addition formula exp(x+y)=exp(x)exp(y) Theorem
The exponential function is smooth and (exp)'=exp Theorem
The exponential function is strictly increasing Theorem

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

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

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)