Taylor Series

Taylor's theorem states that any function satisfying certain conditions can be expressed as a Taylor series.

A polynomial function can be evaluated at any point using simple arithmetic, for example;

$$f(x) = 34 + 4x - 34x^2 + 3x^3$$ is evaluated at some point x=3 like so;

$$f(3) = 34 + (4 \times 3) - (34 \times 3^2) + (3 \times 3^3 )$$ using basic arithmetic operations of addition and multiplication. (e.g. $$3^3$$ is 3 x 3 x 3

A function such as exp, log or sin can be approximated by polynomial functions

this is typically how computers and calculators determine numerical values for log, exp and trig functions in software using some appropriate and efficient approximation to the function.

Polynomial functions are more easily differentiated or integrated, or added or multiplied. So basically polynomial approximations allow complex problems to be simplified into more basic mathematical models.

Taylor polynomials when expressed as an infinite series are known as Taylor series and have some theoretical significance.

Typically after the taylor polynomial is evaluated, then some pattern emerges which can be used to apply the approximation.

Taylor Polynomials
Approximating some function f(a) with its tangent is naturally the best linear approximation to f(a) at that point, and is called a linear Taylor POlynomial about a for f.

The open university course M347 gives the following description of a Taylor Polynomial of degree n about a for some function f as:

$$p_{a,n}(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2 $$

$$  + ... + \frac{f^{(n)} (a)}{n!}(x-a)^n $$

when a=0, the Taylor polynomial simplifies to;

$$p_{0,n}(x)=f(0)+f'(0) x + \frac{f''(0)}{2!} x^2 + ... + \frac{f^{(n)} (0)}{n!} x^n $$

taylor series about 0 are referred to as Maclaurin series

Taylor Polynomial for the exponential function
The taylor polynomial of degree about 0 for $$f(x) = e^x$$

A series series is a Taylor polynomial of infinite degree.

Taylor series are useful when the sequence of Taylor polynomials converges.

$$e^x = 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \frac{1}{4!}x^4 + ... + \frac{1}{n!}x^n, x \in \mathbb{R}$$

$$log(1+x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - \frac{1}{4}x^4 + ... , x \in (-1,1)$$

$$\frac{1}{(1-x)} = x + x^2 + x^3 + x^4 + ... , x \in (-1,1)$$

$$(1 + x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha - 1)}{2!}x^2 + \frac{\alpha(\alpha -1)(\alpha - 2)}{3!}x^3 + \frac{1}{4!}x^4 + ... + \frac{1}{n!}x^n, x \in \mathbb{R}$$

For some functions f(x), the terms go to exactly zero after some cut-off, hence are exactly the same as the series expansion.

To make Taylor series useful, series with converging terms are cut-off after some number of terms depending on the accuracy required.

The approximate form is written;

$$f(x) \simeq f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2 + ... + \frac{f^{r}(a)}{r!}(x-a)^r $$

substituting the variable
let f

adding and subtracting
presumably straight forward

Differentiating and Integrating Taylor Series
@todo c3

variable substitution can be used for example to substitute x with $$x^2$$

arithmetic operations, such as addition by adding the terms

(@todo the exercises in unit 1 2.4.1)