LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-01

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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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A geometric bound for tails of the exponential series

Statement

If $x \in R$ , $N \in N$ , and $ι (N + 2) \geq 2 ∣ x ∣$ , then $\sum_{k = N + 1}^{\infty} \frac{∣ x ∣^{k}}{ι (k!)} \leq \frac{2 ∣ x ∣^{N + 1}}{ι ((N + 1)!)} .$

Facts & Assumptions

Given: $x, N$ with the stated inequality.

[L1]

Factorials satisfy the recurrence; the canonical embedding preserves products and order and is strictly increasing on naturals (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).

[L2]

A geometric tail of ratio $1 / 2$ sums to $2$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

Proof

technique · direct

1.1

For $k \geq N + 1$ , strict increase gives $ι (k + 1) \geq ι (N + 2) \geq 2 ∣ x ∣$ , and the factorial recurrence gives that the ratio of consecutive absolute terms is $∣ x ∣ / ι (k + 1) \leq 1 / 2$ .

givenL1

2.1

Thus the $j$ -th term after $N + 1$ is at most the first tail term times $2^{- j}$ .

step 1.1induction

3.1

Sum the geometric majorant using [L2] to obtain the displayed bound.

step 2.1L2given∎

Depends on

Used by

The elementary numerical bound 2<e<3 Corollary
Checking the symmetric local-lemma condition for a sparse uniform hypergraph Example
The number e is irrational Theorem

Dependency tree · two levels

40 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, Analytic Functions (standard reference, not scraped)
MIT Proofs in Analysis and Probability, Lecture 2 notes (standard reference, not scraped)
LSU MATH 7230, Homework 1 (standard reference, not scraped)