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 $f(x)=g(x)\ \forall x\in U$.

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:

• Taylor series operator is linear
• Taylor series operator commutes with differentiation
• Taylor series operator is multiplicative
• Taylor series operator commutes with composition

Relation with polynomials and power series summation

• Taylor series of polynomial is the same polynomial
• Composite of Taylor series operator and power series summation operator is identity map