Alphabeta MathHow statement and proof provenance work
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Taylor polynomials and their remainders
Definition
Let , let have derivatives through order at , and let be the canonical embedding. The Taylor polynomial of degree at most about is The Taylor remainder is .
The factorials are natural numbers as in The factorial and the falling factorial , defined by recursion in and enter real arithmetic only through (The canonical natural of a field); they are nonzero (Canonical naturals are positive and strictly increasing). The sum and powers are those of Finite sums and finite products, by recursion, Laws of finite sums and finite products, and Integer powers . For , is the constant .
Depends on
- Higher derivatives and the classes $C^k$ and $C^\infty$
- The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$
- Finite sums and finite products, by recursion
- Laws of finite sums and finite products
- Integer powers $a^m$
- The canonical natural $\iota(n) = n \cdot 1_F$ of a field
- Canonical naturals are positive and strictly increasing
Used by
- Taylor and Maclaurin series Definition
- The multivariable Taylor polynomial in multi-index notation Definition
- The Taylor polynomial of (1-x)⁻¹ at 0 has the exact geometric remainder xⁿ⁺¹/(1-x) Example
- Characteristic exponential for a continuous local martingale with deterministic clock Lemma
- Taylor polynomials match the prescribed derivatives at the centre Lemma
- Multidimensional Ito formula for Brownian-driven processes Theorem
- One-dimensional Ito formula Theorem
- Peano's form: the normalized Taylor remainder tends to zero Theorem
- Taylor-series representation by vanishing remainders Theorem
- Taylor's Schlömilch–Roche remainder formula Theorem
- The first nonzero higher derivative classifies a stationary point Theorem
Dependency tree · two levels
39 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 lecture notes (standard reference, not scraped)
- J. Lebl, Basic Analysis I, Taylor's theorem and related calculus (standard reference, not scraped)