Taylor polynomial

Definition

About a general point

Suppose a function $f$ of one variable is defined and at least $n$ times differentiable at a point $x_0$ in its domain. The $n^{th}$ Taylor polynomial for a function $f$ at a point $x_0$ in the domain is the truncation of the Taylor series to powers up to the $n^{th}$ power. If we denote the polynomial by $P_n(f;x_0)$, it is given as:

$P_n(f;x_0)=x\mapsto \sum _{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0{)}^k$

Note that this is a polynomial of degree at most $n$. The degree is exactly $n$ if and only if $f^{(n)}(x_0)\ne 0$.

About the point 0

Suppose a function $f$ of one variable is defined and at least $n$ times differentiable at a point $0$. The $n^{th}$ Taylor polynomial of $f$ at 0 is:

$P_n(f;0)=x\mapsto \sum _{k=0}^n\frac{f^{(k)}(0)}{k!}{x}^k$

Note that this is a polynomial of degree at most $n$. The degree is exactly $n$ if and only if $f^{(n)}(0)\ne 0$.