  It is an open question of Erd\H{o}s as to whether the alternating series
$\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$ is (conditionally) convergent, where
$p_n$ denotes the $n^{\mathrm{th}}$ prime. By using a random sifted model of
the primes recently introduced by Banks, Ford, and the author, as well as
variants of a well known calculation of Gallagher, we show that the answer to
this question is affirmative assuming a suitably strong version of the
Hardy--Littlewood prime tuples conjecture.
