Logrithmic Integral and a Recurrence

It's well known that the logrithmic integral, defined as

                  x
                  /    1
          Li(x) = |  ----- dt
                  /  ln(t)
                 t=2

gives a good approximation for pi(x), the number of primes less 
than x.  Also, we know that the average gap between consecutive 
primes near x is roughly ln(x).  Therefore, the size of the Nth 
prime can be approximated by the Nth term in the sequence s[k] 
whose values is defined recursively by

              s[0] = 2
 
        s[k] = s[k-1] + ln(s[k-1])       for k=1,2,...

Consequently, the value of Li(s[n]) should be close to n.  More
precisely, it appears that

             Li(s[n])   ~=   (n-1)  +-  [error?]

What are the best error bounds on this approximate equality?