Erdős series with alternating prime terms diverges

[Dragonfall]

Erdos noticed that $$\sum(-1)^n\frac{n\log n}{p_n}$$ diverges, where pn is the nth prime. I can't prove this conclusively. All I can say is that PNT implies that p_n~nlogn and thus the series "resembles" $$\sum(-1)^n$$.

 

[Hurkyl]

If the terms don't go to zero, then the sum doesn't converge, right?

 

[Dragonfall]

Oh ya, how the hell did I miss that?