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The Lagrange and Cauchy forms of Taylor's remainder
Statement
Under the hypotheses of Taylor's Schlömilch–Roche remainder formula, there are points between and such that and
Facts & Assumptions
Given: The hypotheses of the Schlömilch-Roche theorem.
For each natural , the Schlömilch-Roche theorem gives a point strictly between and such that (Taylor's Schlömilch–Roche remainder formula).
Factorials and natural powers obey The factorial and the falling factorial , defined by recursion in and Integer powers , while the canonical embedding preserves products and positive naturals are nonzero (The canonical natural of a field, Canonical naturals are positive and strictly increasing).
Proof
Set in [L1]. Then and , giving the Lagrange form.
Set . Since , the formula becomes the Cauchy form.
These are the asserted special cases.
Depends on
Used by
- A uniform derivative bound gives a uniform Taylor remainder bound Corollary
- A bounded fifth derivative gives the central-difference expansion required by Richardson extrapolation Lemma
- A twice differentiable local minimiser has nonnegative second variation Lemma
- MPRK22(1) has a uniform second-order defect and one-step bound on positive tubes Lemma
- Normalized product likelihood expansion under QMD Lemma
- Polyline interpolation and equal-length redistribution approximate a smooth arc-length path Lemma
- Second-order spatial consistency of the periodic FV heat stencil Lemma
- Complex-step differentiation is second-order for holomorphic functions real on the real axis Theorem
- Directional Jacobian and Hessian differences retain Taylor and evaluation-noise errors Theorem
- Forward and central differences have first- and second-order error bounds Theorem
- Multivariable Taylor formula with a Lagrange remainder along a line segment Theorem
Dependency tree · two levels
38 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)
- MathWorld, Schlömilch's remainder (standard reference, not scraped)
- J. Lebl, Basic Analysis I, Taylor's theorem and related calculus (standard reference, not scraped)