View source for 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$f(x)-P_n(x)=o\left((x-x_0)^n\right),\quad x\to x_0,$</td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

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|Taylor formula]].

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