Erdős Problem #15

Is it true that \(\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}\) converges, where \(p_n\) is the sequence of primes? Note: In the problem statement, \(p_n\) is the \(n\)-th prime, indexed such that \(p_1=2, p_2=3, \ldots\). We 0-index here to reflect how Nat.nth works. Note: convergence here is convergence of the sequence of partial sums, which is what the problem asks about. `Summable` would be the wrong notion: it is unconditional summability, equivalent over \(\mathbb{R}\) to absolute convergence, and \(\sum_n n/p_n\) diverges.