DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-14

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.

Taylor and Maclaurin series

Definition

Let $I \subseteq R$ be an open interval, let $f \in C^{\infty} (I)$ (Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ ), and let $a \in I$ . The Taylor series of $f$ at $a$ is the real power series

$\sum_{n = 0}^{\infty} \frac{f^{(n)} (a)}{n!} (x - a)^{n} .$

Its $n$ th partial sum is the Taylor polynomial $T_{n, a} f$ (Taylor polynomials and their remainders). When $a = 0$ , the series is called the Maclaurin series of $f$ . This definition names a power series (A real power series about a centre, its interval of convergence, and its radius in $[0, + \infty]$ ); by itself it asserts neither that the series converges nor that its sum equals $f$ .

Depends on

Used by

A smooth function not equal to its Maclaurin series Counterexample
Standard Maclaurin expansions Theorem
Taylor-series representation by vanishing remainders Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 82 results over 15 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

W. F. Trench, Introduction to Real Analysis, §4.5 (standard reference, not scraped)
J. K. Hunter, An Introduction to Real Analysis, §§10.7.1–10.7.2 (standard reference, not scraped)
MIT OpenCourseWare 18.100C Real Analysis, Lecture 23 (standard reference, not scraped)