Basically the Taylor Series is a function represented as an infinite sum of terms that are ordered like an infinite polynomial.

## Background

Introduced in 1715 by Brook Taylor

- Brook Taylor:
- Born 18 August 1685
- Edmonton, Middlesex, England

- Death 30 November 1731 (aged 46)
- London, England

- English mathematician
- best known for Taylor’s theorem and Taylor series
- attended St. John’s College

- Born 18 August 1685
- Colin Maclaurin:
- Born February, 1698
- Kilmodan, Cowall, Argyll, Scotland

- Death 14 June 1746
- Edinburgh, Scotland

- Scottish mathematician
- Maclaurin Series
- special case of Taylor Series

- Born February, 1698

## The Idea

\[f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+ \cdots\]or for fans of the sigma notation: \[\sum_{n=0} ^ {\infty } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}\]

The core idea here is to be able to write all functions in one common, differientiable and generic way.

## The Concept

Using the Power Series (or the more specific Taylor Series) one can represent complex function as a simple chain of powers, just like in a polynomial. This can drastically improve the way we handle certain functions.

### Power Series

A power series is a simple infinite polynomial: \[f(x) = a_0+a_1 (x-a)+a_2 (x-a)^2+a_3 (x-a)^3 \cdots \hspace{10pt}or\hspace{10pt} \sum_{n=0} ^ {\infty} a_n (x-a)^n\]

The power series is very simple. Every term in the function has some arbitrary coefficient in front that could be anything.

### Taylor Series

Brook Taylor furthermore said that most(every?) functions could be represented as such an power series.

Lets make two assumptions.

- f(x) does in fact have a power representation about x = a
- f(x) has derivatives of every order and we can find them all

At this point we need to find the coefficients \( a_0, a_1, a_2, a_3, …, a_{\infty} \)

Looking back at the original function (\(a_0+a_1 (x-a)+a_2 (x-a)^2+a_3 (x-a)^3 \cdots\)) we can try evaluating it at: \(a\) \[f(a) = a_0\]

Every term will end up going to zero except for the first coefficient.

We can reapply the same idea to the derivatives of the power series. \[\dot{f}(x) = 0+a_1+a_2 2(x-a)+a_3 3(x-a)^2 \cdots\\]

Therefore when plugging in a again: \(f’(a)=a_1\)

For the second derivative: \( \ddot{f}(x) = 0+0+2a_2+a_3 6(x-a) \cdots

\)

A pattern is starting to form here:

\( f^{n}(x) = n!a_n \rightarrow a_n = \frac{f^{(n)}(a)}{n!} \)

**Taylor Series for** \(f(x)\) **about** \(x=a\) \[f(x) = f(a)+\frac {\dot{f}(a)}{1!} (x-a)+ \frac{\ddot{f}(a)}{2!} (x-a)^2+\frac{\dddot{f}(a)}{3!}(x-a)^3+ \cdots = \sum_{n=0} ^ {\infty } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}\]

### Maclaurin Series

If we use \(a = 0\), so we are talking about the Taylor Series about x = 0, we call the series a **Maclaurin Series** for f(x): \[f(x) = f(0)+\frac {\dot{f}(0)}{1!} x+ \frac{\ddot{f}(0)}{2!} (x)^2+\frac{\dddot{f}(0)}{3!}(x)^3+ \cdots = \sum_{n=0} ^ {\infty } \frac {f^{(n)}(0)}{n!} \, (x)^{n}\]

### Examples

**__PLACEHOLDER__**

## Sources

- http://en.wikipedia.org/wiki/Brook_Taylor
- http://en.wikipedia.org/wiki/Colin_Maclaurin
- http://en.wikipedia.org/wiki/Taylor_series