Maclauren series
With a Maclaurin series you can modify functions, to make calculations easier
taylor  = f⁽⁰⁾(0) ∕ 0! + {f⁽¹⁾(0) ∕ 1!}x + {f⁽²⁾(0) ∕ 2!}x² + {f⁽³(0) ∕ 3!}x³+ ⋯ .gif)
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
 = ax³ + bx² + cx + d.gif){kind=small}
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. |