A friendly warning, the following is bit esoteric, (a fancy word for hazy), and is still very much a work in progress. I'm having fun with it though, and I think it's an insightful review of the Taylor series in any event, so if you're interested, read on.
My previous post was about a new way to memorize the Taylor series. It emphasized what the Taylor series did rather than how it was derived. I hadn't completely thought things through and had introduced a new term, "the inverse chain rule". +John Baez pointed out that the explanation as it stood sounded rather mysterious, and he was right. His comment made me rethink the entire Taylor series process again, and I think I, (hopefully), have a much more clear explanation now!
My premise is that the easy way to memorize the Taylor series is to understand what is going on in each term. Hopefully, the same simple thing will be happening in each term making the whole process easy to remember and apply.
First, what does a Taylor series do? It provides an approximation for a function near a point where we already know the function's value. Let's take a look at the first two terms of a Taylor series centered at the point x equals 0:
Let's add one more term to the series
Now, suppose that instead of using a graph I'd decided to use derivatives. My first guess would be that to find the slope of x cubed with respect to x I'd take the third derivative. After all, there's an x cubed in the denominator of the derivative
I'd get the wrong answer though because the third derivative is not the same operation as finding the slope of a function with respect to x cubed. After taking the derivative of x cubed three times, I'd get six, not one, the correct answer I'd found by doing the problem graphically.
Although I can't prove it yet, I claim that this is a general result and so, if I want to get the slope of function with respect to x to the nth power, I use the following formula