A result giving a polynomial approximation of a sufficiently smooth function around a point, together with a remainder term that quantifies the approximation error (several equivalent remainder forms exist, such as the Lagrange and integral forms).

Principle

Local behavior of smooth functions can be captured by finite-degree polynomials; higher-order derivatives control the size of the error term and thus the quality of approximation.

Demonstration

Example: For f(x)=e^x expanded about 0, the n-th degree Taylor polynomial is sum_{k=0}^n x^k/k! and the remainder R_n(x) can be bounded by e^ξ|x|^{n+1}/(n+1)! for some ξ between 0 and x, illustrating concrete error control.

Misapplication

Truncating the series and treating the polynomial as equal to the function outside the radius of convergence or when the function is smooth but non-analytic; this misapplication leads to significant approximation errors.

Consequence

Provides a practical tool for approximating functions, deriving asymptotic expansions, performing numerical computations with error bounds, and proving local qualitative behavior (e.g., classification of critical points via Taylor expansion).

Reversal

If the remainder term is identically zero on an interval then the function equals its Taylor polynomial there, which characterizes polynomials of bounded degree; conversely, approximability by polynomials does not always imply analyticity.

Boundary

Requires differentiability up to order n+1 (for Lagrange remainder) on an interval around the expansion point; distinctions matter between smooth (C^∞) and analytic functions — convergence of the Taylor series to the function is not automatic for smooth functions.

Semantic Tension

Competing meanings arise between a finite Taylor polynomial (an approximation with explicit remainder) and an infinite Taylor series (a power series whose convergence to the function defines analyticity); conflating them causes errors about validity of termwise manipulations and convergence claims.

Synthesis

Taylor's theorem packages local derivative information into a polynomial plus a controlled remainder: derivatives up to order n determine the polynomial approximant and the next derivative bounds the error, unifying approximation, error estimation, and local qualitative analysis.