Analytic on an interval/expressing with a different power series

[Esseintes]

Homework Statement

Suppose the real power series \sum ^{\infty}_{n=0}c_{n}x^{n} has radius of convergence R > 0. Define f:= \sum ^{\infty}_{n=0}c_{n}x^{n} on I:= (-R, R) and let b \in I. Show that there exists a power series \sum d_{n}(x-b)^{n} that converges to f(x) for |x-b| < r - |b|.

Homework Equations

None that I can think of.

The Attempt at a Solution

I don't even know where to begin. Obviously the function is analytic on the open interval I because it is defined by a power series that converges on I. Intuitively, I understand that the function can be represented by a power series with a different center (in I) and smaller radius of convergence, but I can't think of how to start demonstrating this rigorously.

 

[HallsofIvy]

If f(x) is as defined, what can you say about f(u) where u= x+ b? Notice that x= u- b.