Let f be defined by the power series
f(x)=n=0∑∞an(x−1)n,
where the radius of convergence is R>0.
Answer the following questions about f and its series representation.
(a) Explain why f(x) equals its own power–series expansion on the open interval (1−R,1+R).
(b) Suppose F(x)=∫1xf(t)dt and G(x)=f′(x) are obtained by term-by-term integration and differentiation, respectively. Justify why each of the power series for F and G has the same radius of convergence R. Clearly state any theorems or principles you use in your reasoning.