Since the function $f(x)=e^{-x^2}$ cannot be integrated using elementary functions, how could one find a power series for $F$, where $F$ is an elementary function such that $F'(x)=e^{-x^2}$?
2025-01-13 02:47:13.1736736433
Power series for non elementary functions
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Such an $F$ does not exists. There is no anti-derivative that is an elementary function.
Since $e^{-x^2}$ is an analytic function, you can still integrate the power series term by term to get an analytic anti-derivative, but this will not be an elementary function.
(Summarized from the comments.)