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If
, then we can certainly write a Taylor series for . However, analyticity requires that this Taylor series actually converge (at least across some radius of convergence) to . It is not necessary that the power series for converge to , as the following example shows.
Let
Then
, and for any ,
(see below). So the Taylor series for around 0 is 0; since for all , clearly it does not converge to .
Let
be polynomials, and define
Then, for ,
Computing (e.g. by applying L'Hôpital's rule), we see that
.
Define
. Applying the above inductively, we see that we may write
. So
, as required.
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