Consider a divergent series $S$ which can be regularized to a number of convergent ones $S^{n}_\epsilon$ which asymptote to the series $S$ when the regularizing parameter $\epsilon$ is removed.
$$ \forall n: \qquad S^{n}_\epsilon \stackrel{\epsilon \to 0}{\longrightarrow} S \,.$$
We call the divergent series 'asymptotically regular' if there is a finite number $\alpha \lt \infty$ such that
$$ \forall n: \qquad S^{n}_\epsilon = \mathcal O(1/\epsilon) + \alpha +\mathcal O(\epsilon) \,, $$
and we call $\alpha$ the 'finite part of $S$'.
I wish to learn what characterizes the asymptotic regularity of divergences, as in why some of them are asymptotically regular. In that spirit, now consider analytic continuation as a method of regularization.
By some witchcraft, we expect the following conjecture to hold true:
A divergent series S is asymptotically regular and its finite part is $\alpha$ if and only if there is a function $f:D \subset \mathbb R \to \mathbb R$ so that for some $s\notin D: f(s)= S$ in form, and it can be analytically continued to $F:\mathbb C \to \mathbb R$ such that $F(s) = \alpha$.
I am looking for a proof or disproof of the above conjecture. Thank you.