I have two sequences whose relation is described in the following recurrence relations:
$ p_{k + 1} = p_k + \frac{1}{2s_k}$
$ s_{k + 1} = s_k + \frac{s_k}{p_{k+1}}$
(when $p_0=2, s_0 = \frac{1}{2}$ $p_k$ is a probabilistic approximation for the $k+1$th prime number)
I would like to study the asymptotic behavior of the sequences. I have seen the method of generating functions, but this seems to work only if there is one function in the recursive definition. I was able to rewrite some things and get
$ p_{k+1}s_{k+1} = p_ks_k + \frac{1}{2} + s_k $
but this doesn't get me anywhere. Especially, I want to see if $p_k$ approaches $\frac{k}{\log k}$ (like it should do if $p_k$ really was the kth prime number). I was wondering if there is any specific technique or approach to attack this recursive rules.