Abstract
A family of alternating series \begin{equation*} S_{r}\left(u; v_{1}, \ldots, v_{r-1}\right)=\sum_{n \geq 1} \frac{(-1)^{n-1}}{(n+u)^{2} \prod_{j=1}^{r-1}\left(n+v_{j}\right)} \end{equation*} is studied, where$u, v_{j}$are nonnegative integers and the poles are distinct except for the double pole at$-u$. By a classical Euler transform, each series is converted into a polynomial continued fraction, so that every admissible parameter choice yields a structured continued fraction value. An exact rational triple ($\alpha, c, \beta$) is computed symbolically by partial fraction decomposition, and the identity$S_{r}=\alpha \pi^{2}+c \log 2+\beta$is obtained. An exact purity test is thereby obtained, because the presence of the$\log 2$component is decided by the coefficient$c$before numerical computation is performed. For the two-parameter family$S_{2}(u; v)$, a complete purity rule is proved, namely$c=0$if and only if$u$and$v$have the same parity. On the numerical side, continued-fraction convergents are evaluated and are accelerated by Aitken$\Delta^{2}$and Wynn's$\varepsilon$algorithm under a deterministic selection rule. Truncation is controlled by the alternating-series remainder bound, and numerical stability is checked by guard precision and depth variation. Parameter maps and rankings on bounded grids show that, even in the pure region, large differences in practical extraction performance for$\pi^{2}$are produced by different parameter choices.