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In a comment to Evaluating $\int_{0}^{1} \lim_{n \rightarrow \infty} \sum_{k=1}^{4n-2}(-1)^\frac{k^2+k+2}{2} x^{2k-1} dx$ for $n \in \mathbb{N}$ I proposed to study this alternating harmonic sum
$$s(p) = \sum _{k\ge 1} \frac{(-1)^{\binom{k}{p}}}{k}\tag{1}$$
where the sign of the terms is determin...
