# Maclaurin and Taylor series

A Maclaurin series is a power series

$y = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots$

which provides an approximation for y near x=0.

It is a way of “coding” functions into a standard form, i.e. power series.

It doesn’t work for all functions, but it does work for a lot of them.

The coefficient $a_n$ can be calculated from $a_n = \frac{1}{n \cdot (n-1) \cdot (n-2) \ldots (3) \cdot (2)} \frac{d^n y}{dx^n} \lvert _{x=0}$

For many functions the approximation can be made as good as you like, or even as good as you like for x as big as you like, by calculating enough terms of the power series.

Maclaurin series can be used

• to find approximate solutions to differential equations which can’t be solved otherwise
• to calculate values of functions like $\sin x$ for different values of x
• to make it easier to differentiate functions
• to make it possible to integrate functions which can’t be integrated otherwise.

Click here for pdf on Maclaurin series (revised 20/9/17)