I guess this is a simple question. Say I am tasked with finding the Taylor series for a given function. Well say that the function is analytic and so we know there's a taylor series representation for it. Am I gauranteed that this representation is the only power series representation for it? Or could there be others? What if the function is not analytic but we can create a power series representation on some restricted interval? Is that power series unique for it? Also, what if I have a function that I can construct a power series on more than one interval in the domain. How do those series relate to each other?