Difference between revisions of "Uniformly bounded derivatives implies globally analytic"
From Calculus
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Revision as of 14:44, 8 July 2012
Statement
Global statement
Suppose is an infinitely differentiable function on such that, for any fixed , there is a constant (possibly dependent on ) such that for all nonnegative integers , we have:
Then, is a globally analytic function: the Taylor series of about any point in converges to . In particular, the Taylor series of about 0 converges to .
Examples
The functions all fit this description.