is ist posible to apply Borel resummation to sums of the form
$$ 1-2^{s}+3^{s}-4^{s}+....=\eta(-s) $$
of course the idea is to link zeta regularization and Borel summation since
$$ \eta(s)=(1-2^{1-s})\zeta (s) $$
using Borel summastion plus the Theta operator $ \Theta = (x\partial_{x}) $
i think that borel resummation should be equaivalent to Abel resummation and equal to
$$ \eta (-s)= (\Theta )^{s}\frac{x}{1+x}$$