Taylor polynomial

of degree $n$, for a function $f$ that is $n$ times differentiable at $x=x_0$

The polynomial $$P_n(x)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k.$$ The values of the Taylor polynomial and of its derivatives up to order $n$ inclusive at the point $x=x_0$ coincide with the values of the function and of its corresponding derivatives at the same point: $$f^{(k)}(x_0)=P_n^{(k)}(x_0),\quad k=0,\dotsc,n.$$ The Taylor polynomial is the best polynomial approximation of the function $f$ as $x\to x_0$, in the sense that

$f(x)-P_n(x)=o\left((x-x_0)^n\right),\quad x\to x_0,$

(*)

and if some polynomial $Q_n(x)$ of degree not exceeding $n$ has the property that $$f(x)-Q_n(x)=o\left((x-x_0)^m\right),\quad x\to x_0,$$ where $m\ge n$, then it coincides with the Taylor polynomial $P_n(x)$. In other words, the polynomial having the property (*) is unique.

If at least one of the derivatives $f^{(k)}(x)$, $k=0,\dotsc,n$, is not equal to $0$ at the point $x_0$, then the Taylor polynomial is the principal part of the Taylor formula.

Comments

For references see Taylor formula.