Taylor series

Suppose you know everything about a function at one single point a — its value, its slope, its curvature, the rate of change of its curvature, and so on through all the derivatives. The Taylor series is the attempt to reconstruct the entire function from that local data alone, by assembling a power series whose coefficients are exactly those derivatives.

Given a function f that is infinitely differentiable at a, its Taylor series centered at a is sum_{n=0}^infinity f^{(n)}(a)/n! (x - a)^n = f(a) + f'(a)(x-a) + f''(a)/2! (x-a)^2 + ... . The n-th coefficient is the n-th derivative at a divided by n factorial. Each partial sum is a Taylor polynomial, the best polynomial approximation of that degree near a.

Beware a subtle trap: writing down the Taylor series is always possible for a smooth function, but the series need not converge to f, even where it does converge. Whether f equals its Taylor series is governed by the Taylor remainder going to zero, and there exist smooth functions whose Taylor series at a point converges to the wrong answer. A function that does equal its Taylor series near every point is called real-analytic.