I am looking to clarify the relationship between a power series and a Taylor series. I realize that if I discover that a function can be represented by a power series then it seems straight forward I can work out the Taylor series.
First question is: Since I know the radius of convergence of the power series I assume I can apply the same radius to the Taylor Series where the center point would be located in the radius?
Second. If I generate a Taylor series from a function I know that it may not necessarily mean a power series exists for that function but if I find the radius of convergence of the Taylor series does that mean then that a power series does exist for that radius? I realize at least one point of convergence must exist for any power series but is there some other factor that determines if my Taylor series will have a power series associated with it?
Although the main question is question two which is embedded in the title of my post but I feel the first question needs a little clarification for me as well ...Thank you