Taylor series radius of convergence and center

[azay]

When approximating a function with a Taylor series, I understand a series is centered around a given point a, and converges within a certain radius R. Say for a series with center a the interval of convergence is [a-R, a+R].

Does this imply that:

1. There also exists a Taylor series expansion centered in any of the other points in that interval?

2. If not, and one would like to describe the function in terms of Taylor series, in some interval [b-S, b+S] of a point b $$\in$$ [a-R, a+R] and b $$\neq$$ a. And let this interval [b-S, b+S] $$\subset$$ [a-R, a+R]. If there exists no Taylor series expansion centered in b, it is still possible to say something about the interval [b-S, b+S] by looking at the expansion centered in a. But how could one possibly know that a series then does exist in a? How do you find this a?

 

[g_edgar]

1. There also exists a Taylor series expansion centered in any of the other points in that interval?

Yes, centered at any other point of the interval $$(a-R,a+R)$$ . We don't know about the endpoints.