# Contents

## Defintion

Let $f\in {C}^{\infty }\left(ℝ\right)$ a smooth function with $n$th derivative ${f}^{\left(n\right)}\in {C}^{\infty }\left(ℝ\right)$ and let $c$ be a real number.

###### Definition

The Taylor series of $f$ at $c$ is the formal power series

$\sum _{n=0}^{\infty }\frac{1}{n!}{f}^{\left(n\right)}\left(c\right)\left(x-c{\right)}^{n}\phantom{\rule{thinmathspace}{0ex}}.$\sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(c) (x-c)^n \,.
###### Definition

For $f\in {C}^{\infty }\left(ℝ\right)$ a smooth function with $n$th derivative ${f}^{\left(n\right)}\in {C}^{\infty }\left(ℝ\right)$, its Mac Laurin series is its Taylor series at zero:

$\sum _{n=0}^{\infty }\frac{1}{n!}{f}^{\left(n\right)}\left(0\right){x}^{n}\phantom{\rule{thinmathspace}{0ex}}.$\sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(0) x^n \,.
###### Remark

Similarly definition apply to functions on any Cartesian space or smooth manifold.

## Properties

###### Proposition

The Taylor series of a smooth function $f$ at any point may have vanishing radius of convergence, but it is always an asymptotic expansion of $f$.

###### Theorem

(Borel’s theorem)

The map

${C}^{\infty }\left({ℝ}^{k+l}\right)\to {C}^{\infty }\left({ℝ}^{k}\right)\left[\left[{X}_{1},\cdots {X}_{l}\right]\right]$C^\infty(\mathbb{R}^{k+l}) \to C^\infty(\mathbb{R}^k) [ [ X_1, \cdots X_l] ]

obtained by forming Taylor series in $l$ variables is surjective.

In particular, every power series in $ℝ\left[\left[X\right]\right]$ is the taylor series of some smooth function on the real line.

###### Proof

The proof is reproduced for instance in MSIA, I, 1.3

Examples of sequences of infinitesimal and local structures

first order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$←$ differentiationintegration $\to$
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
square-0 ring extensionnilpotent ring extensionring extension
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Revised on November 5, 2013 23:03:38 by Urs Schreiber (82.169.114.243)