How do you write the first 4 nonzero terms AND the general term of the taylor series for #e^((-x)^2)# centered at x=0?
First, we'll take note that
Alright, now there are two ways to find the Taylor series of this function. The first is the standard of calculating the derivatives of
While this would work just fine, it would be time-consuming. There's a much faster way. We can actually make a substitution into a Taylor series that resembles this function which we already know.
What do I mean? Well, imagine just for a second that we have a variable
Clearly, the Taylor series of
Now, you should be able to see that the Taylor series of
(if not, see here. Most calculus students are expected to have memorized some of these basic series)
And there's our answer.