The Terms of $ (-1)^n n/p_n$ Are Not Eventually Monotone

Let $a_n = n/p_n$, where $p_n$ is the $n$-th prime. The alternating series test requires two things of the magnitudes: that they tend to zero, and that they eventually decrease. The first holds. We prove that the second fails on every tail. The criterion is exact and entirely algebraic: $a_n < a_n+1$ precisely when $p_n > n\\,g_n$, where $g_n = p_n+1 - p_n$ is the gap. Clearing denominators turns the comparison of two ratios into a comparison of two products, and the gap appears by substitution. Bounded gaps therefore force a rise as soon as $p_n$ passes $n B$, and since $p_n/n $ every fixed bound $B$ eventually falls below that line. Consequently there is no index beyond which $(a_n)$ decreases, and the convergence of $ (-1)^n a_n$, if it holds, cannot come from the alternating series test.