Sum of alternating reciprocals of logarithm of 2,3,4...

The Alternating Series Test, which is a special case of the Dirichlet Test, ensures the convergence of the first series.

To apply the Dirichlet test to $k/p_k$, one would have to show that the sequence $\{k/p_k\}$ has bounded variation. That is, $$ \sum_{k=1}^\infty\left|\frac{k}{p_k}-\frac{k+1}{p_{k+1}}\right|<\infty\tag{1} $$ I don't know if $(1)$ is true.