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:

$\sum _{k=0}^{\mathrm{\infty }}\frac{f^{(k)}(x_0)}{k!}(x-x_0{)}^k$

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

$f(x_0)+f^{\prime }(x_0)(x-x_0)+\frac{f^{″}(x_0)}{2}(x-x_0{)}^2+\frac{f^{‴}(x_0)}{6}(x-x_0{)}^3+\dots$

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:

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

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

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

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 $C^{\mathrm{\infty }}$-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 $f(x)=g(x)\mathrm{\forall }x\in U$.

Facts

Preservation of structure

Together, these three facts show that the Taylor series operator is a homomorphism of $\mathbb{R}$-algebras that commutes with the differential structure:

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

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