Let f(x)=n=0∑∞an(x−r)n be a power series with positive radius of convergence ρ.
(a) State the domain on which f is known to represent an analytic function and briefly describe how this domain is determined.
(b) Suppose F(x) is defined by the antiderivative formula F(x)=∫rxf(t)dt. Explain why F can be expressed as a power series centered at x=r having the same radius of convergence ρ.
(c) Provide a logical argument, based on an appropriate convergence test, justifying why the radius of convergence of the series for F cannot exceed ρ.
(d) Discuss whether the interval of convergence of the series for F must coincide with that of f, paying special attention to the behavior at the endpoints.
a. Part (a)
b. Part (b)
c. Part (c)
d. Part (d)