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In a Taylor series, the convergence/divergence behavior at the boundary case $|x-x_0|=R$ is not immediately determinable.

If I understand correctly, such distribution of convergence and divergence over the spherical shell can be thought as $S^n\mapsto \hat{\mathbb{C}}$, and this distribution is a function of the original function for which the Taylor series was defined. Is there an extended study done on this subject?

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I don't fully understand your question, but this MO post seems to be relevant:…. – Srivatsan Apr 4 '12 at 4:42
Possible duplicate to… which is also linked to by the MO post Srivatsan mentioned. Given the dearth of literature there probably isn't an "extended" study on the subject. – Willie Wong Apr 4 '12 at 8:21

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