# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 17

*The Taylor formula*

So far, we have only considered power series of the form . Now we allow that the variable may be replaced by a "shifted variable“ , in order to study the local behavior in the *expansion point* . Convergence means, in this case, that some
exists, such that for

the series converges. In this situation, the function, presented by the power series, is again differentiable, and its derivative is given as in Theorem 16.1 . For a convergent power series

the polynomials yield polynomial approximations for the function in the point . Moreover, the function is arbitrarily often differentiable in , and the higher derivatives in the point can be read of from the power series directly, namely

We consider now the question whether we can find, starting with a differentiable function of sufficiently high order, approximating polynomials
(or a power series).
This is the content of the *Taylor expansion*.

## Definition

Let denote an interval,

an -times differentiable function, and . Then

*Taylor polynomial of degree*for in the point .

So

is the constant approximation,

is the linear approximation,

is the quadratic approximation,

is the approximation of degree , etc. The Taylor polynomial of degree is the (uniquely determined) polynomial of degree with the property that its derivatives and the derivatives of at coincide up to order .

## Theorem

Let denote a real interval,

an -times differentiable function, and an inner point of the interval. Then for every point , there exists some such that

and .

### Proof

## Corollary

Suppose that is a bounded closed interval,

is an -times continuously differentiable function, an inner point, and . Then, between and the -th Taylor polynomial, we have the estimate

### Proof

The number exists due to Theorem 11.13 , since the -th derivative is continuous on the compact interval . The statement follows, therefore, directly from Theorem 17.2 .

*Criteria for extrema*

We have seen in the 15th lecture that it is a necessary condition for a differentiable function to have a local extremum at a point, that its derivative equals at this point. We give now an important sufficient criterion, which relies on higher derivatives.

## Theorem

Let denote a real interval,

an -times continuously differentiable function, and an inner point of the interval. Suppose that

- If is even, then does not have a local extremum in .
- Suppose that is odd. In case , the function has an isolated local minimum in .
- Suppose that is odd. In case , the function has an isolated local maximum in .

### Proof

Under the given conditions, the Taylor formula becomes

with some
(depending on )
between
and .
Depending on whether
or
holds, we have
(due to the continuity of the -th derivative)
or
for
,
for a suitable
.
For these , we have
,
so that the sign of depends on the sign of .

For even, is odd and therefore the sign of changes at
(for
,
the sign is negative, and for
,
the sign is positive).
Since the sign of does not change, the sign of is changing. This means that there can not be an extremum.

Suppose now that is odd. Then is even, hence
for all
in the neighborhood. This means, in the neighborhood, in case
,
that
holds, and we have an
isolated minimum
in . If
,
then
holds, and we have an
isolated maximum
in .

A special case of this is that in case
and
,
then we have an isolated minimum, and in case
and
we have an isolated maximum.

*The Taylor series*

## Definition

Let denote an interval,

an infinitely often differentiable function, and . Then

*Taylor series*of in the point .

## Theorem

Let denote a power series which converges on the interval , and let

denote the function defined via Theorem 12.2 . Then is infinitely often differentiable, and the Taylor series of in coincides with the given power series.

### Proof

That is infinitely often differentiable, follows directly from Theorem 16.1 by induction. Therefore, the Taylor series exists in particular in the point . Hence, we only have to show that the -th derivative has as its value. But this follows also from Theorem 16.1 .

## Example

We consider the function

given by

We claim that this function is infinitely often differentiable, which is only in not directly clear. We first show, by induction, that all derivatives of have the form with certain polynomials , and that therefore the limit for equals (see Exercise 17.16 and Exercise 17.17 ). Therefore, the limit exists for all derivatives and is . So all derivatives in have value , and therefore the Taylor series in is just the zero series. However, the Function is in no neighborhood of the zero function, since .

*Power series ansatz*

Taylor series/R/Power series/Section

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