Taylor Polynomial

Core Sentence / 核心句

A Taylor polynomial is a finite-degree polynomial whose derivatives match a function’s derivatives at a chosen center.

中文理解：Taylor polynomial 用有限个 polynomial terms，在中心点逐阶匹配原函数的 value、slope、curvature 和更高阶导数。

P_{n} (x) = k = 0 \sum n \frac{f ^{(k)} ( a )}{k !} (x - a)^{k}

Matching condition：

P_{n}^{(k)} (a) = f^{(k)} (a), 0 \leq k \leq n

看到 approximate near $a$ 、degree $n$ 、small-angle approximation、estimate function value，就考虑 Taylor polynomial。

Taylor polynomial 是 finite approximation，不是 infinite Taylor series；离 center 越远通常越要关注 truncation error。