Real Analysis 45 | Taylor's Theorem [dark version]

Taylor’s theorem turns local smoothness into a controlled approximation: if a function has enough derivatives, its value near a chosen expansion point can be written as a polynomial whose coefficients come from derivatives at that point, plus a remainder term that shrinks predictably as the input gets close. That remainder behavior is what makes the approximation more than a heuristic—linearization and higher-order polynomial approximations come with a quantitative error structure.

The setup starts by fixing an expansion point x0 and looking at nearby inputs x. Writing x as x0 + H (with H small) makes the formulas cleaner. The simplest case is the derivative definition: the best linear approximation takes the form f(x0 + H) = f(x0) + f'(x0)H + R(H)·H, where the remainder factor satisfies R(H) → 0 as H → 0. In other words, the tangent line captures the function up to first order, and the leftover term becomes negligible compared to H.

Taylor’s theorem generalizes this idea to higher degrees. With enough derivatives, the function admits an nth-order Taylor polynomial:

f(x0 + H) = Σ_{k=0}^{n} [f^{(k)}(x0)/k!] · H^k + R_n(H).

The polynomial part has degree at most n because the powers of H run from H^0 up to H^n. The coefficients are determined by the derivatives of f at x0, scaled by factorials (so the quadratic term becomes (1/2!)f''(x0)H^2, the cubic term (1/3!)f'''(x0)H^3, and so on). The remainder term R_n(H) accounts for what the polynomial misses.

Crucially, the remainder can be expressed using an (n+1)st derivative evaluated at an intermediate point C. For H such that x0 + H stays inside the interval I, there exists C between x0 and x0 + H with

R_n(H) = [f^{(n+1)}(C)/(n+1)!] · H^{n+1}.

This “between points” condition matters: even though C is not known explicitly, its location guarantees the estimate needed for error bounds. The theorem also admits a compact big-O form when the exact constant is not the focus: writing f(x0 + H) = (Taylor polynomial) + O(H^{n+1}) signals that the neglected part scales like H^{n+1} as H → 0.

Finally, the same statement can be rewritten in terms of x directly instead of H, using x − x0 in place of H. Either way, the takeaway is consistent: Taylor polynomials provide increasingly accurate local approximations, and the remainder’s order—n+1—tells exactly how fast the error shrinks as the input approaches the expansion point. That structure underpins a wide range of applications, from numerical methods to analysis of function behavior.