Let the zeta function be defined as:
$\zeta_{A}^{B}(x) = \sum_{n=A}^{B} \frac{1}{n^{x}}$,
where $A,B \in \mathbb{Z}$ with $A\leq B$ and $x \in \mathbb{C}$.
Consider now $a_{n} = 1, 2, 4, 11, 31, 83, 227,...$ (OEIS-A002387) which is based on $\zeta_{1}^{\infty}(1)$.
Then numerical evidence indicates that:
$ e^{x-1} = \lim_{n \rightarrow \infty} ~ \frac{ \zeta_{A}^{B}(x) }{ \zeta_{B}^{C}(x) } ~~~~~~~~~~~(\text{with}~A=a_{n+0}, ~B=a_{n+1} ~\text{and}~ C=a_{n+2})$,
for all $x \in \mathbb{C}$ except those $x$ of the form $1+2ki\pi$ with $k\in\mathbb{Z}\backslash\{0\}$.
My question: Is this conjecture known or even proven/disproven ?