# Numb3rs: Taylor Series

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
• Colin Maclaurin:
• Born February, 1698
• Kilmodan, Cowall, Argyll, Scotland
• Death 14 June 1746
• Edinburgh, Scotland
• Scottish mathematician
• Maclaurin Series
• special case of Taylor Series

## 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.

1. f(x) does in fact have a power representation about x = a
2. 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): $f(x) = f(0)+\frac {\dot{f}(0)}{1!} x+ \frac{\ddot{f}(0)}{2!} (x)^2+\frac{\dddot{f}(0)}{3!}(x)^3+ \cdots = \sum_{n=0} ^ {\infty } \frac {f^{(n)}(0)}{n!} \, (x)^{n}$

__PLACEHOLDER__

## Sources

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

## The Numb3rs Series

Numb3rs used to be its own subsection on this site, where I wrote math & science tutorials. Unfortunately this never came to frution so I integrated ths articles with the rest of the blog. If you want to check out the other Numb3rs posts, you can get an overview here.

## By Cecil Wöbker I do science during the day and develop or design at night. If you like my work, hire me.

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