According to WolframAlpha partial sums for
(I actually used the Maple notation for the input)
sum(prime(n)/prime(n+1)/n^2,n=1...infinity)
seems to converge value-wise (looking at partial sums) in the vicinity of 1.12984...
Are there any references to such calculations using actual (known by now) prime numbers?
Of course we have the trivial upper bound
$$
\sum_{n=1}^{\infty}\frac{p_n}{p_{n+1}n^2}<\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}.
$$
Hence the sum converges.
For better estimates for the sum we could use
$$
\log n+\log \log n-1<{\frac {p_{n}}{n}}<\log n+\log \log n\quad {\text{for }}n\geq 6.
$$
or even better estimates, see here.
