Many a times it is not discussed as to How the Taylor/Maclaurin series came to be in its current form. This short snippet is all about it.
Let us assume that some function can be written as a power series expansion. i.e
We are left with the task of finding out the coefficients of the power series expansion.
Substitution x = 0, we obtain the value of .
Lets differentiate wrt x.
Evaluating at x =0 , we get
That’s it we have found all the coefficient values, the only thing left to do is to plug it back into the power series expression:
The above series expanded about the point x = 0 is called as the ‘Maclaurin Series’. The same underlying principle can be extended for expanding about any other point as well i.e ‘Taylor Series’.