Taylor series

The Taylor series of a smooth function $f$ (at a given point $c$) is a formal power series (in $x - c$) whose partial sum?s 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.

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.

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

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

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:

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

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

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 $f(x)$.

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:

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

**(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).

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

**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 | 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*

Revised on March 7, 2014 08:58:23
by Toby Bartels
(64.89.53.137)