# Taylor remainder theorem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Under sufficient differentiability, a Lagrange remainder and derivative bound control the Taylor approximation error.

mko-taylor-remainder-theorem · theorem · v0.10.0



## 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

[An exact Lagrange remainder for a cubic](https://ocme.evemisslab.com/en/derivations/derivation-mko-taylor-remainder-theorem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Taylor polynomial](https://ocme.evemisslab.com/en/mko/mko-taylor-polynomial/) — Use the finite center jet.
- [Higher-order derivative](https://ocme.evemisslab.com/en/mko/mko-higher-order-derivative/) — Require the next derivative across the interval.
- [Mean value theorem](https://ocme.evemisslab.com/en/mko/mko-mean-value-theorem/) — Retain the Lagrange-remainder background.
- [Absolute value](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/taylor-remainder-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-taylor-remainder-theorem/index.md)