Taylor polynomial

The polynomial formed from the first k derivatives of f at a point a.

Taylor polynomial

Let $f$ be a real- (or complex-) valued function defined on an interval containing $a\in\mathbb{R}$ , and assume that $f^{(j)}(a)$ exists for $0\le j\le k$ (see higher derivatives ). The Taylor polynomial of degree $k$ of $f$ at $a$ is

T_k f(x;a) := \sum_{j=0}^k \frac{f^{(j)}(a)}{j!}(x-a)^j.

Taylor polynomials provide the canonical local polynomial approximation to $f$ near $a$ . Taylor’s theorem quantifies the error via a remainder term .

Examples:

For $f(x)=e^x$ , $T_2 f(x;0)=1+x+\frac{x^2}{2}$ .
For $f(x)=\sin x$ , $T_3 f(x;0)=x-\frac{x^3}{3!}$ .
For $f(x)=\frac{1}{1-x}$ , the Taylor polynomial at $0$ is $T_k f(x;0)=\sum_{j=0}^k x^j$ .