I am interested in the series comprising the inverses of the products of twin prime pairs: $$\sum_i \frac{1}{p_iq_i}$$ where $p_i=6i-1,\ q_i=6i+1;\ (p_i,q_i) \in \mathbb P$. This series is equivalent to $$\sum_k \left(\frac{1}{36k^2-1}\right)$$ for $k \in$ OEIS 002822. The term $\frac{1}{3\cdot 5}$ for the twin prime pair that does not have the desired form is omitted from the series as I consider it, but it can readily added to the result if doing so would be in any way enlightening.
Computationally, the first forty terms sum to $0.0412994859\dots$ In this range, the final terms are $\approx 10^{-6}$
The series converges. I offer two proofs. First: $$p_i^2<p_i\cdot p_{i+1} \Rightarrow \frac{1}{p_i^2}>\frac{1}{p_i\cdot p_{i+1}} \Rightarrow \sum_i \left(\frac{1}{p_i^2}\right) > \sum_i \left(\frac{1}{p_i\cdot p_{i+1}}\right)$$ But $\sum_i \left(\frac{1}{p_i^2}\right)=P(2)$ where $P(2)$ is the prime zeta function, which converges. So the sum of the inverse of the products of all pairs of sequential primes must converge, and the series that I am interested in is a subseries thereof, and hence also converges.
Second: The sum of the inverses of the primes which occur in twin primes converges to Brun's constant. But for each twin prime pair, the sum of the inverses $\frac{1}{p}+\frac{1}{p+2}=\frac{2p+2}{p(p+2)}$ is much larger than the inverse of the product itself $\frac{1}{p(p+2)}$. Hence the series I am interested in must converge.
Despite much thought and some searching, I have not been able to find relevant literature or progress beyond this point.
My questions are: (1) Has my series been examined previously, and if so, where might I locate those studies? (2) Can anyone provide me with some methods or insights as to how to evaluate the series $\sum_i \frac{1}{p_iq_i}$?