Let be defined by the power series f(x)= _ n=0 ^ a_n ,(x-1)^n
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
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.