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