# Taylor polynomial

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

At center a, derivative values of orders0 through n, scaled by their factorials, form the order-n Taylor polynomial.

mko-taylor-polynomial · definition · v0.10.0



## Mathematical statement

Fix center a and nonnegative order n and require every f^(k)(a) for0≤k≤n to exist. T_n(x)=sum over k=0..n of f^(k)(a)*(x−a)^k/k!; at n=0 this is constant f(a). The polynomial itself gives no remainder bound or guarantee that an infinite Taylor series converges or equals f. A polynomial f of degree at most n reconstructs exactly; this does not make an arbitrary smooth function analytic.





## Representative formula (exact TeX)

T(f,a,n)





## Assumptions


- Positive families supply a complete derivative jet or exact rational/integer polynomial coefficients in the centered power basis.
- Factorial division uses exact rational arithmetic, not floating approximations or an omitted center shift.





## Symbol scope


- T — Taylor-polynomial operator
- f — function with a finite derivative jet
- a — expansion center
- n — nonnegative order





## Intuition, definition, and misconceptions

Coefficients are f^(k)(a)/k!, not unscaled derivative values.

The centered basis is (x−a)^k.

A finite polynomial supplies no automatic remainder or analyticity conclusion.

Do not omit factorials, move a jet to a different center or identify the finite polynomial with an arbitrary original function or infinite series.





## Adopted derivation guide

[A Taylor polynomial retaining the center and factorials](https://ocme.evemisslab.com/en/derivations/derivation-mko-taylor-polynomial-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Differentiating polynomial sums, products, and compositions](https://ocme.evemisslab.com/en/mko/mko-derivative-rules/) — Use derivative values and polynomial differentiation.
- [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/mko/mko-polynomial/) — The finite Taylor sum is an algebraic object.
- [Factorial](https://ocme.evemisslab.com/en/mko/mko-factorial/) — k! scales each coefficient.
- [Higher-order derivative](https://ocme.evemisslab.com/en/mko/mko-higher-order-derivative/) — Every derivative order from0 through n is required.





## Verification boundary

This slice constructs finite polynomials only; remainder, convergence and analyticity require separate proofs.

not_formalized

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





## Next obligation

Add the Taylor remainder theorem.





## 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-polynomial.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-taylor-polynomial/index.md)