If the generating function summation and zeta regularized sum of a divergent series exist, do they always coincide?

Just an idea.
The zeta-function has a laurent-series-representation as a sum of the term $$\zeta_a(x)=-{1\over1-x}$$ and of the series $$\zeta_b(x) = 1/2 + 0.081x-0.0031x^2+... = \sum_{k=0}^\infty c_k x^k$$ such that $$\zeta (x)=\zeta_a(x)+\zeta_b(x)$$ If I recall correctly, the series $\zeta_b(x)$ is entire - this means, that in the case of arguments $x$ where the zeta becomes divergent, the divergent part is completely eaten by the $\zeta_a(x)$ part - and the behave of the geometric series is completely accepted as regular even in the divergent cases by the expression as fraction ${1\over1-x}$ with the sole exception of the case where $x=1$.

This is not a proof for the identity of the zeta-regularization and the power-series/generating function, but it shows at least a strong formal relation ship: the zeta -even in the divergent cases as formal sum of two powerseries regularly computable- and the power series of the generating function.
I think someone with more formal background might be able to put this two things together.