# Analytic on an interval/expressing with a different power series

1. Apr 1, 2009

### Esseintes

1. The problem statement, all variables and given/known data

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|.

2. Relevant equations

None that I can think of.

3. 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.

Last edited: Apr 1, 2009
2. Apr 2, 2009

### HallsofIvy

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

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