Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 1 / 12F / a / iv / Solution

The termwise differentiation of a power series theorem states that inside its radius of convergence, the sum is differentiable and its derivative is the sum of the differentiated series. Accordingly $\boxed{f^{\prime }(x)=g(x)\text{ for }|x|<R}$. There is no blanket differentiability assertion at the convergence endpoints.