The series is convergent for the and for example, hence we can expect that the radius of convergence is greater than . Now you have to show it's indeed .
This is from a practice exam:
Part (a) is fairly straightforward. Let and . Then , which means we can define by , where and are the restrictions of and to and , respectively. It follows immediately that is analytic on .Let have a radius of convergence 2, and let have a radius of convergence 2. Let .
(a) If for , prove there is an analytic function for which when and when .
(b) For the function of part (a), determine the radius of convergence for the power series .
(c) What relationship exists between the coefficient sequence and the coefficient sequence ?
However, I'm lost for parts (b) and (c). Any help would be much appreciated!