For any $s \in \mathbb{C}$ with \Re(s)>0, denote by $\eta_{n-1}(s)$ the $(n-1)^{th}$ partial sum of the Dirichlet series for the eta function $\eta(s)=1-2^{-s}+3^{-s}-\cdots \;$, and by $R_n(s)$ the corresponding remainder. Denoting by $u_n(s)$ the segment starting at $\eta_{n-1}(s)$ and ending at $\eta_n(s)$, we first show how, for sufficiently large $n$ values, the circle of diameter $u_{n+2}(s)$ lies strictly inside the circle of diameter $u_n(s)$, to then derive the asymptotic relationship $R_n(s) \sim (-1)^{n-1}/n^s$, as $n \rightarrow \infty$. Denoting by D=\left\{s \in \mathbb{C}: \; 0< \Re(s) < \frac{1}{2}\right\} the open left half of the critical strip, define for all $s\in D$ the ratio $\chi_n^{\pm}(s) = \eta_n(1-s) / \eta_n(s)$. We then prove that the limit $L(s)=\lim_{N(s)