This is for real analysis so I'm not worried about complex analytic functions.
The definition in my book just says: "A function f(x) which is represented by a power series with a positive radius of convergence is said to be 'real analytic at the origin', or simply 'analytic'."
To me it seems like this definition ends prematurely. Like when I was reading it I expected there to be an "if" and then a list of qualifications.
I guess I just don't really understand the rationale behind creating this definition in the first place. What would even be the point of representing a function by a power series that didn't have a positive radius of convergence? If the power series doesn't converge for any values of x, it's not really representative of anything, right?
I feel like this is actually obvious and I'm just overthinking it, but is this definition literally just giving a name to functions that can be represented as power series that actually converge for x values in some radius? There was an exercise earlier in the chapter before analytic functions were defined that asked us to find an infinite power series which represented a function; was that function analytic as well?
Thanks in advance and sorry if this is a silly question.