Series

[Dragonfall]

1. The problem statement, all variables and given/known data

Prove that \sum(-1)^n\frac{n}{p_n} converges, where p_n is the nth prime.

2. Relevant equations

The sequence \frac{n}{p_n} is definately not monotone if there exists infinitely many twin primes, since 2n-p_n<0 for sufficiently large n, so alternating series test is out. Are there any other ways of showing this converges?

[StatusX]

Can you use the prime number theorem? This says that:

\lim_{n \rightarrow \infty} \frac{p_n}{n \ln n} = 1

[Dragonfall]

I can't use it for the series. I can only establish that n/p_n -> 0, which is insufficient for the series. I can't even prove that n/p_n is NOT monotone for large n, unless I assume the twin prime conjecture, for example.