- #1

- 43

- 4

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

(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?