### Maclaurin series

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

##### Explanation

We take the cubic function

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

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

##### Additional information

The Scottish mathematician Colin Maclaurin is best known for the Maclaurin series.