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 $f^{(n)}(x_{0})\neq 0$.

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 $f^{(n)}(0)\neq 0$.