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Erland

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If a function is analytic in a symmetric neighbourhood of a point a, then the Taylor series about a is the only power series about a which represents the function in this neighbourhood. Because if we differentiate the power series repeated times and evaluate at a, we find that the coefficients of the series are determined by the derivatives of the function at a.

And if there is a power series representation about a in such a neighbourhood, then the function is analytic there (it could be taken as a definition of "analytic") and the series is the Taylor series about a.

And if there is a power series representation about a in such a neighbourhood, then the function is analytic there (it could be taken as a definition of "analytic") and the series is the Taylor series about a.

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