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added an Idea-paragraph to non-perturbative effect, also to perturbation theory - divergence/convergence.
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Strictly speaking, the $n$-th derivatives for the function $g \mapsto e^{-1/g^2}$ don’t exist at $g = 0$, because $g \mapsto -1/g^2$ and thus $g \mapsto e^{-1/g^2}$ is only defined for $g \gt 0$ or $g \lt 0$. One could analytically continue $g \mapsto e^{-1/g^2}$ and its $n$-th derivatives to $g = 0$ via a piecewise function
$g \mapsto \begin{cases} e^{-1/g^2} & g \gt 0 \vee g \lt 0 \\ 0 & g = 0 \end{cases}$and similarly for its $n$-th derivatives, but in that case, it shouldn’t be surprised that this function’s taylor series expansion at zero is only valid at zero. It’s the same reason why the taylor series expansion for the absolute value function as the identity function on the real numbers is only valid for the positive real numbers, and is inherently due to the piecewise nature of the function.
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