Analytic functions are usually defined to be those functions which can be represented by their Taylor series. This is what I meant by `tautological'.
In the complex plane (and all real analytic functions can be analytically extended into the complex plane), you can define the equivalent property as saying a (first!) derivative has the some value no matter how you chose to take the limit.
So just by virtue of having a well defined first derivative in a particular region you get all the Maclaurin series magic.