Taylor polynomial

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

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 — Use derivative values and polynomial differentiation.
• Univariate polynomials over the rationals — The finite Taylor sum is an algebraic object.
• Factorial — k! scales each coefficient.
• 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.