Taylor series

Definition

About a general point

Suppose $f$ is a function that is infinitely differentiable at a point $x_{0}$ in its domain. The Taylor series of $f$ about $x_{0}$ is the power series given as follows:

$\sum _{k=0}^{\infty }{\frac {f^{(k)}(x_{0})}{k!}}(x-x_{0})^{k}$

Here's a version with the first few terms written explicitly:

$f(x_{0})+f'(x_{0})(x-x_{0})+{\frac {f''(x_{0})}{2}}(x-x_{0})^{2}+{\frac {f'''(x_{0})}{6}}(x-x_{0})^{3}+\dots$

About the point 0

In the special case of the above definition where $x_{0}=0$ (and in particular $f$ is infinitely differentiable at 0), the Taylor series is as follows:

$\sum _{k=0}^{\infty }{\frac {f^{(k)}(0)}{k!}}x^{k}$

Here's a version with the first few terms written explicitly:

$f(0)+f'(0)x+{\frac {f''(0)}{2}}x^{2}+{\frac {f'''(0)}{6}}x^{3}+\dots +$

Well defined on germs of a functions

The Taylor series operator about a point $x_{0}$ can be thought of as a mapping:

(Germs of $C^{\infty }$ -functions defined about $x_{0}$ ) $\to$ (Formal power series centered at $x_{0}$ )

In fact, this mapping is a $\mathbb {R}$ -algebra homomorphism that commutes with the differential structure.

Here, two functions $f$ and $g$ are said to have the same germ about a point $x_{0}$ if there is an open interval $U$ containing $x_{0}$ such that $f(x)=g(x)\ \forall x\in U$ .

Relation with Taylor polynomials

The Taylor series can be viewed as a limit of Taylor polynomials. 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 $f^{(n)}(x_{0})\neq 0$ .

Whether or not the Taylor series of a function converges to the function is determined by whether or not the sequence of Taylor polynomials of the function converges to the function.

Facts

Preservation of structure

Together, the first three facts show that the Taylor series operator is a homomorphism of $\mathbb {R}$ -algebras that commutes with the differential structure. The fourth fact show that it preserves an additional structure: