Analysis WebNotes
We shall show that the power function xa has a power series expansion about x0=1. (We shouldn't expect that this function have a power series expansion centered on x0=0, since for general real a, xa isn't even defined for x=0.)

First, let's consider what the power series for xa could possibly be. If it is in fact true that xa has a power series expression, then by Proposition 7.22, xa must be equal to its Taylor Series. So let's compute the Taylor Series centered at x=1 for this function.

Of course computing the Taylor Series in no way proves that xa is equal to this series. All it establishes is that this is the only power series which xa could possibly be equal to. But but now we have something something concrete to work with, and we can use other tricks to prove the two functions are equal .