# Taylor's Theorem

## Idea

The Taylor polynomials of a differentiable function approximate it by polynomial functions; the various versions of Taylor’s Theorem describe how good this approximation is. The limiting case of this is the Taylor series.

## Preliminary definitions

Let $f$ be a partial function on a cartesian space $\mathbb{R}^d$ (or $\mathbb{C}^d$), let $c$ be a point in the cartesian space, and let $k$ be a natural number (actually we can allow $k \geq -1$ also, using negative thinking).

###### Definition

If $f$ is differentiable $k$ times at $c$, then the Taylor polynomial of $f$ at $c$ with order $k$ is the unique polynomial in $d$ variables of degree at most $k$ whose derivatives at $c$ match those of $f$ up to order $k$.

It is straightforward (by differentiating a polynomial ansatz?) to find an explicit formula (in the variable $x$):

$T^k_{f@c}(x) = \sum_{n = 0}^k \frac{f^{(n)}(c) (x-c)^n}{n!} .$

We have written this as if $f$ is a function of one variable; but interpret $n$ as a multi-index? whose length $\ell$ satisfies $0 \leq \ell \leq k$, with $f^{(n)}$ the mixed partial derivative given by that multi-index, $n!$ the factorial of $\ell$, and $(x-c)^n$ a product of $\ell$ factors of the form $x_i - c_i$ (where $i$ is an index appearing in the multi-index $n$). Then this works in any cartesian space. (Note that we include all possible orderings of a multi-index as separate terms; this leads to repeated terms that, when combined, will cancel some of the factors in the factorial.)

In the case of a function of several variables, we can also manage the maximum degrees in the various variables separately, although nobody seems to bother with this.

## Statements

There are several different versions of Taylor's Theorem, all stating an extent to which a Taylor polynomial of $f$ at $c$, when evaluated at $x$, approximates $f(x)$.

Here is the simplest statement, which requires only continuity of $f$ (which we really only need for $k = 0$, since it's automatic for $k \geq 1$ and not actually necessary for $k = -1$):

###### Theorem

If $f$ is continuous at $c$, then

$\lim_{x \to c} \frac{{f(x) - T^k_{f@c}(x)}}{{\|x - c\|}^k} = 0 .$

(The norm ${\|\cdot\|}$ can be left out in one variable, or placed in the numerator to handle all components of a vector-valued function at once.)

If $f^{(k)}$ has some continuity, then we get a version of Taylor's Theorem with an integral:

###### Theorem

If $f^{(k)}$ is absolutely continuous on $[\min(x,c),\max(x,c)]$, then

$f(x) = T^k_{f@c}(x) + \int_{t=a}^x \frac{f^{(k+1)}(t) (x-t)^{k+1} \,\mathrm{d}t}{(k + 1)!} .$

(Note that $f^{(k+1)}$ is defined almost everywhere and Lebesgue integrable, because $f^{(k)}$ is absolutely continuous.)

This result is not directly very useful is one is using Taylor polynomials to approximate $f$ where one doesn't know its behaviour, but we have a corollary which can often be used:

###### Theorem

If $f^{(k)}$ is absolutely continuous on $[\min(x,c),\max(x,c)]$, then

${|f(x) - T^k_{f@c}(x)}| \leq \frac{M {|x-c|}^{k+1}}{(k + 1)!} ,$

where $M$ is any essential upper bound of $f^{(k+1)}$ on $[\min(x,c),\max(x,c)]$.

In many cases, finding a good upper bound of $f^{(k+1)}$ can be reduced to solving $f^{(k+2)}(t) = 0$.

There are also versions that generalize the mean value theorem:

###### Theorem

If $f^{(k)}$ is differentiable on $[\min(x,c),\max(x,c)]$, then, for some $t \in [\min(x,c),\max(x,c)]$,

$f(x) = T^k_{f@c}(x) + \frac{f^{(k+1)}(t) (x-c)^{k+1}}{(k + 1)!} .$

All of these can be generalized in a fairly straightforward way to functions of several variables.

Last revised on March 9, 2014 at 10:53:47. See the history of this page for a list of all contributions to it.