Taylor series

In analysis using named after Brook Taylor Taylor series to represent a smooth function in the neighborhood of a point by a power series. The series is the limit of the Taylor polynomials and Taylor series expansion is called this development.

• 3.1 Exponential Functions and Logarithms
• 3.2 Trigonometric Functions

• 4.1 Example

• 5.1 radius of convergence 0
• 5.2 A function that can not be developed into a Taylor series in a development site

• 6.1 Example

Definition

Be an open interval, a smooth function and an element of. Then is called the infinite series

The Taylor series of development with location. Herein the faculty of and the -th derivative of, with the emphasis.

The series is only "formal " means here first, that is, it is not assumed that it converges. In fact, there are Taylor series which converge ( see above for illustration) everywhere.

In the special case of the Taylor series is also called Maclaurin series.

The sum of the first two terms of the Taylor series

Also called linearization in place. Generally referred to as the partial sum

Which is fixed for a polynomial in the variables, the -th Taylor polynomial.

The Taylor formula with remainder term makes statements about how this differs from the polynomial function. Due to the simplicity of the polynomial and the good applicability of the remainder formulas are Taylor polynomials a commonly used tool of analysis, numerical analysis, physics and engineering sciences.

Properties

The Taylor series for the function is a power series with the leads:

And thus by induction

Agreement on the development site

Because of

Determines the Taylor series and their derivatives at the development site with the function or its derivatives match:

Equality with the function

In the case of an analytic function, Taylor series with this power series is consistent, because

And thus.

Major Taylor series

Exponential functions and logarithms

The natural exponential function is shown on all the way through their Taylor series with development point 0:

When the natural logarithm has the Taylor series with radius of convergence 1 development site 1, ie for the logarithm function is represented by its Taylor series (see figure above):

As the following series converges faster

It is suitable for practical applications.

If one chooses a for, then and.

Trigonometric Functions

For the development site ( Maclaurin series) applies:

Here is the -th Bernoulli number and the -th Euler number.

Product of Taylor series

The Taylor series of a product of two real functions and can be calculated if the derivatives of these functions are known at the identical development site:

Using the product rule then results

If the Taylor series of the functions given explicitly

So is

With

This corresponds to the Cauchy product formula of two power series.

Example

Be, and. Then

And we get

In both cases,

And thus

This Taylor expansion would of course also directly on the calculations of the derivatives of possible:

Taylor series are not analytic functions

The fact that the Taylor series at each development site has a positive radius of convergence and convergence in their area with matches, does not apply to any infinitely differentiable functions. But also in the following cases of analytic functions the associated power series is called the Taylor series.

Radius of convergence 0

The function

Is on all infinitely differentiable, but its Taylor series is

And thus only convergent ( ie against or equal to 1).

A function which can not be developed into a Taylor series at a developing position

The Taylor series of a function does not converge always to the function. In the following example, the Taylor series outvoted in any environment around the development site with the output function match:

As a real function is as often continuously differentiable, the derivatives at each point ( for particular ) are all 0. The Taylor series around the zero point is thus the zero function, and is consistent in any neighborhood of 0 with. It is therefore not analytical. The Taylor series converges to a development point between 0 and against. Even with a Laurent series can not approximate this function, because the Laurent series, which represents the function for correctly, results for non- constant 0.

Multidimensional Taylor series

Let now the following is a infinitely often continuously differentiable function with development site. Then the function evaluation can represent as for any, where

If you calculate from the Taylor expansion at the point of development and applies them in, so you get the merhrdimensionale Taylor developing its own:

With the multidimensional chain rule and the multi- index notations

We obtain further that

With the notation we obtain for the multi-dimensional Taylor series with respect to the development point

The one-dimensional case, if you use the multi - index notation to match.

Advertised sees the multi-dimensional Taylor series as follows:

Example

For example, valid according to the set of black for the Taylor series of a function that depends on, on the development site: