/sci/ - Science & Math

Let's say I have a function described by some power series. Borel's theorem says that every power series is the Taylor series of some function. Yet, a lot of the time these power series have a limited range of convergence. I can only ever see a small fraction of the actual function. To "shift" the window over on the function requires you to know the derivatives of that function (as every point of the function corresponds to a separate Taylor series parameterized by the function's derivatives). At that point you might as well already have an expression of the function, which defeats the point.

With all that being said, is there some general method that would allow me to "shift the window of convergence" around on a power function, so that I can see more of the function it describes?