The sums above are Riemann sums over different partitions for the function $\arctan(x)$ between $x=0$ and $x=1$, so the limit will equal the integral which is $\frac{\pi}{4}$. Both of these converge to the same value because the mesh of these partitions goes to zero.
Remark: When I state that the mesh goes to zero, for the primes I am using the result, originally due to Hoheisel, that there exists $\theta<1$ with $p_n-p_{n-1}\ll p_n^\theta$. The best known value is $\theta=0.525$ due to Baker, Harman and Pintz.