I ask if the series $$\sum_{k=1}^{\infty}\frac{p_{k+1}-p_k}{(p_{k+1}+p_k)^\alpha}$$ where $p_k$ stands for the prime of index $k$, has the same properties of convergence of the series $$\sum_{k=1}^{\infty}\frac{1}{k^\alpha}$$ that is convergent for all $\alpha \gt 1$ and divergent for all $\alpha \le 1$.

In the case $\alpha = 1$, I conjecture the following asymptotic behavior of the sum of the series $$\sum_{k=1}^{n}\frac{p_{k+1}-p_k}{p_{k+1}+p_k}\sim \gamma \log n$$ while in the case $\alpha = 2$ the series seems to converge to the value $0.1200307...$