nLab Taylor series

Idea

The Taylor series of a smooth function $f$ (at a given point $c$) is a formal power series (in $x - c$) whose partial sums are the Taylor polynomials of $f$ (at $c$). As the Taylor polynomials are approximations of $f$ by polynomials (up to a given degree), so the Taylor series is an approximation of $f$ by an analytic function (or at least an asymptotic expansion that attempts to be this).

See also Taylor's theorem for error estimates in the convergence of Taylor series.

Definition

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

Definition

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

$$\sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(c) (x-c)^n \,.$$

Definition

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

$$\sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(0) x^n \,.$$

Remark

Similar definitions apply to functions on any Cartesian space or smooth manifold.

Properties

Recall that a (partial) function $f$ is analytic at $c$ (in the interior of $\dom f$) if there exists a power series $P$ at $c$ and a neighbourhood $U$ of $c$ such that, for all $x \in U \cap \dom f$, $P(x)$ converges to $f(x)$.

Proposition

If $f$ is analytic at $c$, then the only power series witnessing this is the Taylor series of $f$ at $c$ (so in particular, the Taylor series exists; analytic functions are smooth).

In contrast, a smooth function need not be analytic; the classic counterexample is a bump function. In fact, the Taylor series of $f$ at $c$ might not converge to $f$ anywhere except at $c$, either because the Taylor series has vanishing radius of convergence or because it converges to something else (an analytic function with the same jet as $f$ but a different germ). However, we can say this:

Proposition

(Taylor series is asymptotic series)

The Taylor series of a smooth function $f$ is always an asymptotic expansion of $f$.

This follows from the Hadamard lemma, see this exampleeries#TaylorSeriesOfSmoothFunctionIsAsymptoticSeries) for details.

Theorem

(Borel’s theorem)

The map

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

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

In particular, every power series in one real variable is the Taylor series of some smooth function on the real line (even if it has vanishing radius of convergence and so is not the Taylor series of any analytic function).

Proof

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

See more at Borel's theorem.

Examples

• of the natural logarithm: see Mercator series

Related concepts

• series, asymptotic series

• analytic geometry

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

• analogue in (stable) homotopy theory/Goodwillie calculus: Taylor tower