Taylor remainder theorem

Mathematical statement

Declare center a, point x, order n and the complete polynomial coefficients, then compute the Taylor polynomial and exact remainder directly. Degree≤n gives zero remainder. For degree n+1 the next derivative is constant, and a strictly intermediate midpoint supplies an exact Lagrange witness when x differs from a. Boundary families accept a qualified M at least the exact derivative supremum D or an undersized witness0≤M<D, rejecting negative M and unknown modes. A finite center jet does not prove the general remainder theorem, equality with a Taylor series or analyticity.

Representative formula (exact TeX)

R(f,a,x,n)=r

Assumptions

• Polynomial instances use exact rational arithmetic.
• The general theorem retains interval-wide differentiability and a derivative bound as explicit obligations.

Symbol scope

• R — Taylor-remainder operator
• f — differentiable function
• a — expansion center
• x — evaluation point
• n — Taylor order
• r — exact remainder

Intuition, definition, and misconceptions

Compute the full Taylor polynomial and remainder.

Separate zero-remainder and exact degree-n+1 witnesses.

Distinguish the derivative supremum from an undersized proposed bound.

A finite center jet or a few samples do not establish the general remainder theorem or analyticity.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Taylor polynomial — Use the finite center jet.
• Higher-order derivative — Require the next derivative across the interval.
• Mean value theorem — Retain the Lagrange-remainder background.
• Absolute value — Use absolute error bounds.

Verification boundary

Exact polynomial cases certify only the specified model. The actual bound inequality may accidentally hold at x=a even with undersized M; this is not a qualified derivative bound.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add analytic Taylor-series convergence.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.