To be concrete, I would like to limit my question to the exponential function and real numbers. The power series centered at any point converges to the exponential because its radius of convergence is infinite. Yet, when calculating the Taylor series about two different centers (say 0 and 1) the coefficients of the Taylor series are not the same. How is it possible then that both series expansions converge to the same exponential? Shouldn't the coefficients be the same? Is it possible to operate with the expansions about the two centers and prove that they are actually the same polynomial, namely that the coefficients after reordering do are the same?
2026-03-31 03:33:50.1774928030
Coefficients of taylor series of an analytic function about different centers.
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