nLab
Taylor series (changes)

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Defintion

For $f \in C^{∞} (ℝ)$ a smooth function with $n$ th derivative $f^{(n)} \in C^{∞} (ℝ)$ and $c$ a real number, its Taylor series at $c$ is the power series

\sum_{n = 0}^{∞} \frac{1}{n!} f^{(n)} (c) (x - c)^{n}

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

\sum_{n = 0}^{∞} \frac{1}{n!} f^{(n)} (0) x^{n}

Similarly for functions on any Cartesian space or smooth manifold.

(Borel’s theorem)

The morphism

C^{∞} (ℝ^{k + l}) \to C^{∞} (ℝ^{k}) [[X_{1}, \dots X_{l}]]

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

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

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	finite
	$\leftarrow$ differentiation		integration $\to$
derivative		Taylor series		germ	smooth function
tangent vector		jet		germ of curve	curve
Lie algebra		formal group		local Lie group	Lie group
Poisson manifold		formal deformation quantization		local strict deformation quantization	strict deformation quantization

Revised on February 6, 2013 18:32:11 by Urs Schreiber (82.113.106.234)