Is there any way to determine if a function is periodic or not from the taylor series expression of it?
Elementarily,it can be easily argued that the series must be an alternating one,but that does not guarantee it's periodicity.Is there any trick?
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See also https://math.stackexchange.com/questions/63102/how-to-prove-periodicity-of-sinx-or-cosx-starting-from-the-taylor-seri and https://math.stackexchange.com/questions/1029266/is-it-possible-to-detect-periodicity-of-an-analytic-function-from-its-taylor-ser – Miguel Aug 07 '17 at 11:41
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If $f$ is analytic, then it is $T$-periodic iff $\forall k, \ f^{(k)}(0) = \sum_n (2 i \pi n T)^k c_n$ where $c_n = \frac{1}{T}\int_0^T f(x) e^{-2i \pi nx}dx$ – reuns Aug 07 '17 at 11:53