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

And likewise:

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’.

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