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 they are not too weirdly distributed among $[0,1]$.

**Remark:** We need to use the fact that there exists $\theta<1$ with $p_n-p_{n-1}\ll p_n^\theta$.  (we can take $\theta=0.525$)  For the primes this tells us that if $j\geq n^{0.525+\epsilon}$, then
$$p_{n+j}-p_{n}\sim j\log n. $$

**Edit:**  I added why $p_n-p_{n-1}\ll n^{0.525}$ is important.  It tells us/(or actually comes from) how things will look in short intervals.  It is not true that for general sequences with $\alpha_{i}-\alpha_{i-1}\ll n^{-\delta}$ the Riemann sum works out, rather for sequences where sums over short intervals is very close to the identity function.