### Maclauren series

With a **Maclaurin series** you can modify functions, to make calculations easier

taylor

##### Explanation

The fundamental theorem of mathematics says that a function can always be made in the form

Here we take the cubic function that we write as

Subsequent derivatives of this function give

There is some regularity in this. The coefficients of the terms develop still further, will eventually become zero and disappear. That is clearly visible at the point *x* = 0 where you get

Here we see a factorial, and write it as

which gives

, , ,

The coefficients of the original function are determined by the derivatives of this function at the point 0. Substitution gives

We change the sequence of the terms in

This applies to a cubic function. We say very frankly, that this also means

and it turns out to be true.

##### Example 1

The exponential function has for every value of *x* the series development

## HistoryThe Scottish mathematician Colin Maclaurin (1698 - 1746) is best known for the Maclaurin series. |