Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is(adsbygoogle = window.adsbygoogle || []).push({});

$$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n \,\,\,\,\,\,\,\,\,\,\,\,\,\forall x \in (x_0-r,x_0+r) $$

(In particular, I do not know in advance anything about the analyticity of ##f(x)## in ##(a,b)##, but only the convergence of the Taylor series in ##x_0##.)

Does this imply that, chosen a ##x_1 \in (x_0-r,x_0+r)##, the Taylor series of ##f(x)## centered in ##x_1## converges at least in ##(x_1-r',x_1+r')## with ##r'=r-|x_0-x_1|##? That is

$$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_1)}{n!} (x-x_1)^n \,\,\,\,\,\,\,\,\,\,\,\,\,\forall x \in (x_1-r',x_1+r') $$

If this is true, how can I prove it?

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# I Convergence of Taylor series in a point implies analyticity

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