Erdős Problem #15 — Convergence of the alternating series $\sum(-1)^n n/p_n$

I just discovered in it a forgotten conjecture of mine, which might still be of interest. Let $\{p_n\}$ be the sequence of consecutive primes. Is it true that $\sum_{n=1}^{\infty}(-1)^n(n/p_n)$ converges? If not, what happens to the partial sums?