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Mathematics
Calculus
Convergence of Taylor series in a point implies analyticity
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[QUOTE="stevendaryl, post: 5640896, member: 372855"] Fooling around with the power series, I see that the conclusion is easily provable if you additionally know the following: [LIST=1] [*]The series for [itex]f(x)[/itex] is absolutely convergent (so that you can reorder terms and get the same answer). [*]The series for [itex]\frac{df}{dx}, \frac{d^2 f}{dx^2}, ...[/itex] are also absolutely convergent. [/LIST] As to point two--can you prove that if [itex]f(x)[/itex] is equal to its Taylor series at a point, then [itex]f^{(n)}(x)[/itex] is equal to its Taylor series at that point? As to point one--I'm not sure whether you can get away without assuming that, or not. [/QUOTE]
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Mathematics
Calculus
Convergence of Taylor series in a point implies analyticity
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