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Taylor's theorem

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 ff be a partial function on a cartesian space d\mathbb{R}^d (or d\mathbb{C}^d), let cc be a point in the cartesian space, and let kk be a natural number (actually we can allow k1k \geq -1 also, using negative thinking).

Definition

If ff is differentiable kk times at cc, then the Taylor polynomial of ff at cc with order kk is the unique polynomial in dd variables of degree at most kk whose derivatives at cc match those of ff up to order kk.

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

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

We have written this as if ff is a function of one variable; but interpret nn as a multi-index? whose length \ell satisfies 0k0 \leq \ell \leq k, with f (n)f^{(n)} the mixed partial derivative given by that multi-index, n!n! the factorial of \ell, and (xc) n(x-c)^n a product of \ell factors of the form x ic ix_i - c_i (where ii is an index appearing in the multi-index nn). 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 ff at cc, when evaluated at xx, approximates f(x)f(x).

Here is the simplest statement, which requires only continuity of ff (which we really only need for k=0k = 0, since it's automatic for k1k \geq 1 and not actually necessary for k=1k = -1):

Theorem

If ff is continuous at cc, then

lim xcf(x)T f@c k(x)xc k=0. \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)f^{(k)} has some continuity, then we get a version of Taylor's Theorem with an integral:

Theorem

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

f(x)=T f@c k(x)+ t=a xf (k+1)(t)(xt) k+1dt(k+1)!. 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)f^{(k+1)} is defined almost everywhere and Lebesgue integrable, because f (k)f^{(k)} is absolutely continuous.)

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

Theorem

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

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

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

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

There are also versions that generalize the mean value theorem:

Theorem

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

f(x)=T f@c k(x)+f (k+1)(t)(xc) k+1(k+1)!. 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.