Taylor polynomial

Definition

About a general point

Suppose $n$ is a nonnegative integer. 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 $f^{(n)}(0)\neq 0$.

Intuition

The $n^{th}$ Taylor polynomial, intuitively, is an attempt to be the best local approximation of $f$ about $x_{0}$ among polynomials of degree $\leq n$.

Particular cases

Value of $n$	$n^{th}$ Taylor polynomial about $x_{0}$	What it means	$n^{th}$ Taylor polynomial about $x_{0}$ case $x_{0}=0$
0	$f(x_{0})$	This is a constant function whose value is the value of $f$ at $x_{0}$. Clearly, this is the best approximation for $f$ among approximations by constant functions.	$f(0)$
1	$f(x_{0})+f'(x_{0})(x-x_{0})$	The graph of the function is a straight line that equals the tangent line to the graph of $f$ at $(x_{0},f(x_{0}))$. The tangent line is intuitively the best linear approximation of the graph of the function, so this makes sense.	$f(0)+f'(0)x$.