Let $p(n)$ be the $n$th prime, $g(n) = p(n+1) - p(n)$ be the $n$th prime gap, and $r(n) = g(n) / p(n)$ be the $n$th relative prime gap. The Prime Number Theorem implies that $r \rightarrow 0$.
Consider the series $$\sum_n (-1)^n r(n).$$ This is an alternating series with terms going to 0, albeit not monotonically. Morally (assuming no conspiracy of primes) and computationally it seems this sum converges to 0. Is it known whether this series does indeed converge and if so, whether the limit is 0?