I refer to my previous question Asymptotic behavior of a certain sum of ratios of consecutives primes. We can split the sum $$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$ where $p_k$ stands for the prime of index $k$, into the following two

$\sum_{k=1}^{n}\frac{p_{k+1}}{p_{k+1}-\,p_k}$ ~ $\frac{n\,(n+1)}{e}\,\log\log n$

$\sum_{k=1}^{n}\frac{p_{k}}{p_{k+1}-\,p_k}$ ~ $\frac{(n-1)\,n}{e}\,\log\log n$

Is there anybody who can confirm this asymptotic behavior and, if it is correct, give a sketch of a proof?