Brun's Theorem given in $1919$ ensures that the sum:
$$B_{2}=∑_{p,p+2}\left(\frac1p+\frac1{p+2}\right)≃1.90216054...$$
of the reciprocals of the twin primes converges ($d=2$).
My question is: Is the sum:
$$∑_{p,p+d}\left(\frac1p+\frac1{p+d}\right)$$
of the reciprocals of the primes spaced by $d=2k$ converges for all $k≥1$.
Yes, Brun’s sieve can no doubt achieve this. But we can actually say a bit more, namely that the number of primes $p\le x$ such that $p+d$ is at most $O(x/(\log x)^2)$ for any $d>1$ (with the implied constant depending on $d$), and this implies convergence of the reciprocal series.
Many expositions of the Selberg sieve will prove this upper bound as an application. (You could search for “twin primes” with “Selberg sieve” and find several writeups: you will find that other values of $d$ only change the argument at finitely many primes which does not qualitatively affect the result.